,transcriber's ,note3 ,italics are indicated by underscores4 ,,the ,,engineer ,,series ,,the ,,gyroscopic ,,compass ,a ,,non-,,mathematical ,,treatment ,,the ,,engineer ,,series ,,the ,,gyroscopic ,,compass ,a ,,non-,,mathematical ,,treatment ,,by ,t4 ,w4 ,,chalmers1 ,b4,,sc41 ,a4,m4,i4,,mech4,e4 7,,on ,,the ,,editorial ,,staff ,,of 8,,the ,,engineer07 ,,illustrated ,,london ,,constable ,,company1 ,,ltd4 #aj ,,orange ,,street1 ,,leicester ,,square4 ,w4,c4 #aibj ,printed in ,great ,britain ,,preface ,the chapters composing this book originally appeared as a series of articles in ,the ,engineer during ,january1 ,february1 and ,march of the current year4 ,the articles were written in the belief that many readers would welcome a clear and full1 non-mathematical exposition of the gyroscopic compass1 its theory and practical construction4 ,the gyro-compass represents at once the most involved and abstruse and the most important and valuable of all the practical applications to which the gyroscope1 so far1 has been put4 ,as a navigational instrument it is now in practically universal use in all the chief war navies of the world1 and is to-day being adopted by several important representatives of the mercantile marine4 ,remarkable figures were shown to the author recently which demonstrated that not only was navigation by the gyro-compass much more accurate than by the magnetic compass1 but that the increased accuracy reduced the length of the voyage of a mercantile vessel to an extent that resulted in saving a quantity of fuel the value of which on a single trip would go a considerable way towards meeting the extra first cost of the gyro-compass4 ,bearing these facts in mind the author from the outset endeavoured not only to dispense with mathematics but to avoid introducing anything except the most familiar physical principles and conceptions1 for his object was to explain the mode of action of the gyro-compass for the benefit primarily of the navigating officer--naval and mercantile4 ,if some readers should find the treatment in places unduly prolix1 the author trusts they will exercise leniency and regard the fault as being caused by the author's unwillingness to take any risks in expounding a subject1 no part of which can be understood incompletely without grave hurt to the understanding of the whole4 ,t4 ,w4 ,c4 ,,london1 ,may1 #aibj4 ,,contents ,,chap4 ,,page ,i4 ,,introduction #a ,,ii4 ,,elementary ,,gyroscopic ,,phenomena #d ,,iii4 ,,the ,,gyroscope ,,and ,,the ,,rotation ,,of ,,the ,,earth #ae ,,iv4 ,,damping ,,the ,,vibrations ,,of ,,the ,,gyro-,,compass #bi ,v4 ,,the ,,damping ,,system ,,of ,,the ,,ansch,,tz 7#aiaj7 ,,compass #db ,,vi4 ,,the ,,damping ,,system ,,of ,,the ,,sperry ,,compass #eb ,,vii4 ,,the ,,damping ,,system ,,of ,,the ,,brown ,,compass #ei ,,viii4 ,,the ,,latitude ,,error #fe ,,ix4 ,,the ,,north ,,steaming ,,error #ga ,x4 ,,the ,,ballistic ,,deflection #ha ,,xi4 ,,the ,,quadrantal ,,error #ia ,,xii4 ,,the ,,elimination ,,of ,,the ,,quadrantal ,,error #ajg ,,xiii4 ,,centrifugal ,,forces ,,during ,,quadrantal ,,rolling #acj ,,xiv4 ,,the ,,ansch,,tz 7#aiaj7 ,,compass #ach ,,xv4 ,,the ,,sperry ,,compass #adb ,,xvi4 ,,the ,,brown ,,compass #adh ,,xvii4 ,,the ,,ansch,,tz 7#aiab7 ,,compass #aed ,,index #afe ,,list ,,of ,,illustrations ,,fig4 ,,page #a4 ,model gyroscope with three degrees of freedom #d #b4 ,model gyroscope with three degrees of freedom #g #c4 ,model gyroscope1 frictional transmission of turning moment #i #d4 ,model gyroscope1 one degree of freedom lost #aa #e4 ,model gyroscope1 second degree of freedom lost #ab #f4 ,model gyroscope1 lost degrees of freedom restored #ac #g4 ,elementary gyroscope at equator #ae #h4 ,gyroscopic clock #af #i4 ,elementary gyro-compass #ah #aj4 ,elementary compass at equator #bj #aa4 ,elementary compass at #ee deg4 ,n4 ,lat4 #bd #ab4 ,compass at equator and near ,north ,pole #bf #ac4 ,pendulum and compass #cb #ad4 ,damped and undamped vibrations #ce #ae4 ,damped pendulum #cg #af4 ,air-blast damping system of ,anschtz 7#aiaj7 compass #dc #ag4 ,free and damped motion of axle #di #ah4 ,damping curve from ,anschtz 7#aiaj7 compass #ej #ai4 ,damping system of ,sperry compass #ec #bj4 ,action of excentric pin in ,sperry compass #ee #ba4 ,action of excentric pin in ,sperry compass #eg #bb4 ,gyro-pendulum with axle tilted #fj #bc4 ,damping system of ,brown compass #fb #bd4 ,the north steaming error at #j deg4 and #fj deg4 ,n4 #gb #be4 ,sperry correction mechanism for latitude and north steaming errors #gf #bf4 ,ballistic force on compass when ship's speed changes #hc #bg4 ,ballistic force on compass when ship's speed changes #hd #bh4 ,ballistic deflection #hf #bi4 ,effect of rolling on due north course #ic #cj4 ,effect of rolling on due west course #id #ca4 ,external gimbal mounting #ig #cb4 ,effect of rolling on a due north course 7simple mounting7 #ih #cc4 ,effect of rolling on a due north course 7external gimbal mounting7 #ii #cd4 ,ship rolling on ,n4,w4 course #aja #ce4 ,sperry compass on ,n4,w4 course #ajh #cf4 ,sperry ballistic gyro #aaa #cg4 ,stabilised excentric pin 7,sperry compass7 #aab #ch4 ,diagram of ,brown compass #aac #ci4 ,oil control bottles 7,brown compass7 #aae #dj4 ,brown compass on west course #aag #da4 ,diagram of ,anschtz 7#aiab7 compass #aba #db4 ,plans of gyros 7,anschtz compass7 #abb #dc4 ,centrifugal forces on a pendulum #aca #dd4 ,the ,anschtz 7#aiaj7 compass #aci #de4 ,the ,sperry compass removed from binnacle #adc #df4 ,the ,sperry compass #add #dg4 ,the ,brown compass removed from binnacle #adh #dh4 ,the ,brown compass removed from binnacle #adi #di4 ,the ,brown compass #aej #ej4 ,plan of ,anschtz 7#aiab7 compass #aee #ea4 ,sectional elevation of ,anschtz 7#aiab7 compass #aei ,,the ,,gyroscopic ,,compass3 ,a ,,non-,,mathematical ,,treatment ,,chapter ,i ,,introduction ,at this date it is1 or should be1 unnecessary to open an account of the gyroscopic compass with a discussion of the defects of the ordinary magnetic compass4 ,these defects are too well known to require mention4 ,recent advances in naval architecture1 particularly in warship construction1 and very especially the building of submarines1 have resulted in the magnetic compass becoming less and less useful for accurate navigation1 primarily because of the upsetting influence exercised upon it by masses of steel or iron in its neighbourhood4 ,it may still serve1 perhaps1 for the surface navigation of submarines1 but for submerged runs the use of a gyro-compass is all but essential4 ,in warships the weight of the guns and turrets is now so heavy that the magnetic compass can hardly remain unaffected by them and is materially influenced when the guns are trained to different directions4 ,the shells themselves as they are discharged are also said to be a cause of error in the reading of the magnetic compass1 for they tend in most positions of the ship to drag the needle after them by magnetic attraction as they pass along the bore of the gun4 ,the value of the gyro-compass is not1 however1 recognised only in the world's war navies4 ,it is becoming increasingly appreciated in the mercantile marine1 and there can be but little doubt that the device will soon be extensively employed on passenger liners and merchantmen generally4 ,in the following pages we attempt to give an account of the working of the gyro-compass and to describe the forms assumed by the device in practice--sufficiently fully to illustrate the theory without going into any great detail on the constructional side--and to do so without depending upon the reader's possessing mathematical knowledge4 ,it is to be remarked that it is much easier to treat the gyroscope and all its practical applications mathematically than non-mathematically1 and that the avoidance of mathematics generally leads to a discussion of this essentially mathematical device which is unscientific1 unsound1 and of very little practical value4 ,we trust that our account will be found to avoid these defects and that it will prove useful and enlightening to those who have so far failed to understand the behaviour of the gyroscope and its applications by reason of the fact that hitherto all trustworthy descriptions have been couched in a highly mathematical form or have been mere mathematics thinly disguised in written words4 ,it is admittedly not easy to understand gyroscopic phenomena either with or without the aid of mathematics1 but on the other hand many of the difficulties of the subject are largely artificial4 ,thus the mathematician1 when dealing with it1 seems to be much more concerned with his equations than in creating a mental picture of what they represent2 yet every one of his equations can be or should be capable of being represented physically4 ,those who set out to avoid mathematics do not usually succeed in giving a discussion sufficiently complete to be of any practical service afterwards to their readers4 ,thus in dealing with the gyro-compass the so-called 8popular0 description in most cases begins and ends with an explanation of why the device possesses directive force when it is set up at the equator4 ,it is quite easy to demonstrate the existence of such force at the equator4 ,it is not so easy to show non-mathematically how the directive force is generated and applied when the compass is situated north or south of the equator4 ,the necessity for damping the horizontal vibrations of the gyro-axle and how the required damping force is applied in practice are still more difficult to explain1 while the errors to which the gyro-compass is open--such as the latitude and the quadrantal errors--are even more trying to make clear4 ,the latter subjects are usually neglected in the 8popular0 account of the compass4 ,yet without some means of damping the vibrations referred to or of eliminating or allowing for the various errors1 the compass1 even though it can be shown to possess directive force in all latitudes1 would be utterly useless--especially on board ship--as a direction indicator4 ,finally1 it may be remarked that while the gyro-compass represents to-day probably the most intricate and involved practical application of the gyroscope1 it is not the only one of importance4 ,this fact is to our advantage1 for if we succeed in explaining the theory and working of the gyro-compass we shall have succeeded in placing the reader in a position enabling him readily to understand all other devices in which a gyroscope is employed or in which gyroscopic phenomena are developed4 ,,chapter ,,ii ,,elementary ,,gyroscopic ,,phenomena ,let a wheel ,a 7,fig4 #a7 be mounted on an axle ,b ,c journalled within a horizontal ring ,d4 ,let this ring in turn be mounted on journals ,e ,f within a vertical ring ,g and1 further1 let this vertical ring be carried on journals ,h ,j within a vertical frame ,k4 ,this arrangement constitutes a gyroscopic system having three degrees of freedom1 because relatively to the frame ,k the wheel may turn about three axes ,b ,c1 ,e ,f1 and ,h ,j mutually at right angles to each other and because1 if the wheel is set spinning on its axle1 gyroscopic properties will be manifested4 ,illustration3 ,,fig4 #a4 ,model ,gyroscope1 with ,three ,degrees of ,freedom4 ,the following is a brief statement of the gyroscopic properties manifested when the wheel is spun on its axle3 7a7 ,let the wheel be spinning in the direction of the arrow ,l and let a weight ,w be hung on the horizontal ring at the end ,b of the axle4 ,the movement produced by this weight is not a rotation of the horizontal ring1 and the wheel within it1 about the axis ,e ,f4 ,instead1 the horizontal ring remains horizontal and the whole system inside the square frame sets off rotating at a uniform speed about the axis ,h ,j in the direction of the arrow marked ,m on the horizontal ring4 ,this rotation or precession1 as it is called1 will be maintained so long as the weight ,w remains in action4 ,there is here no question of perpetual motion4 ,the work expended in overcoming the friction at the vertical journals is derived from the energy of the spinning wheel1 and when this energy is exhausted the phenomenon ceases4 ,the phenomenon can1 in fact1 only be maintained indefinitely by expending power to drive the wheel against the leakage of energy through friction at the journals of the axle and the vertical axis ,h ,j4 ,a closer examination of the phenomenon would show that there is a slight rocking motion of the horizontal ring on its axis ,e ,f1 and therefore an additional leakage of energy at the journals of this axis4 ,this rocking motion can be neglected for our present purposes4 ,it is sufficiently accurate to say that the horizontal ring remains horizontal4 7b7 ,the speed of the precession is proportional to the weight ,w and to the speed of rotation of the wheel on its axle4 ,for instance1 doubling the weight doubles the speed of precession4 7c7 ,if the direction of spin of the wheel is reversed the direction of the precession is also reversed4 7d7 ,if the spin of the wheel is in the direction ,l1 and if instead of attaching a weight at the end ,b of the axle we exert an upward force at this point the precession developed will be opposed to the direction of the arrow ,m4 7e7 ,if instead of trying to rotate the wheel about the axis ,e ,f by means of a weight or force applied at ,b we attempt to turn it about the vertical axis ,h ,j by applying a horizontal force ,v to the outer ring1 the wheel will not turn about the vertical axis ,h ,j1 but about the horizontal axis ,e ,f1 the end ,b of the axle rising up towards ,h4 7f7 ,as before1 the direction of this movement is reversed by reversing either the direction of spin of the wheel or action of the force ,v4 ,if both are reversed simultaneously the direction of the movement produced by the applied force is not altered4 ,the behaviour set forth above can be summarised in a general rule as follows3--,if to a spinning wheel possessing three degrees of freedom a force be applied tending to turn the wheel about some axis ,x ,x1 the actual motion produced will not be about ,x ,x but about some other axis ,y ,y2 this axis ,y ,y will be such that rotation about it will tend to bring the axle of the spinning wheel into coincidence with or parallel with the axis ,x ,x2 the direction of the rotation produced about ,y ,y will be such that when the condition of coincidence or parallelism is reached the spin of the wheel will coincide in direction with the rotation we are attempting to produce about the axis ,x ,x4 ,taking case 7a7 7,fig4 #a71 it will be seen that the axis ,e ,f about which we are attempting to produce rotation by means of the weight ,w1 together with the weight ,w itself1 is of necessity carried round by the precession in the direction ,m at the same rate as the axle of the spinning wheel4 ,the axle in this case cannot therefore place itself in coincidence with the axis of the applied force4 ,but it does its best to do so4 ,the precession persists and is an expression of the fruitless chase of the axis ,e ,f by the axle ,b ,c4 ,illustration3 ,,fig4 #b4 ,model ,gyroscope1 with ,three ,degrees of ,freedom4 ,if1 however1 the weight is attached by some kind of sliding connection on the horizontal ring in such a way that its line of action remains stationary in space1 then the axis about which we are attempting to produce rotation will also remain stationary in the position occupied by the axis ,e ,f before precession commences4 ,in this case it is quite possible for the axle of the wheel to place itself in coincidence with the axis of the applied force4 ,precession about ,h ,j through #ij deg4 will accomplish this result1 as indicated in ,fig4 #b4 ,the weight ,w is now acting at a point on the horizontal ring where it ceases to have any tendency to turn the wheel about the axis ,e ,f4 ,when1 therefore1 the position of coincidence is reached precession ceases and the system comes to rest in this position4 ,if the experiment were actually made it would be found that the momentum acquired by the system during the #ij deg4 turn would carry the axle through the position of coincidence with the axis of the applied force4 ,but immediately the axle passes to the opposite side the force ,w is exerted on a point of the horizontal ring between ,f and ,c4 ,the action of the force passing on to this1 the opposite1 segment of the ring reverses the conditions under which the system started its movement and as a result precession in the direction opposed to the arrow ,m is set up4 ,the axle thus tends to recover its position of coincidence and in the end settles down to a vibratory motion from side to side of the axis of the applied weight4 ,friction at the vertical journals will 8damp0 this vibratory motion1 the amplitudes of the swings will decrease1 and the axle will ultimately settle in steady coincidence with the axis of the applied force4 ,in this condition the force will have no further effect on the system beyond throwing a bending moment on to the vertical axis4 ,illustration3 ,,fig4 #c4 ,frictional ,transmission of ,turning ,moment4 ,instead of trying to make the wheel rotate about the axis ,e ,f by applying a weight to the inner ring as in ,fig4 #a1 let us1 as shown in ,fig4 #c1 mount the square frame ,k on a horizontal axis ,n ,p and attach the weight ,w to an arm fixed on the frame4 ,the axes ,n ,p and ,e ,f being--at least initially--collinear1 the effect of this arrangement is to throw a turning moment on to the wheel about the axis ,e ,f just as does the weight ,w in ,fig4 #a4 ,it is to be noticed1 however1 that the moment of the weight ,w in ,fig4 #c about the axis ,n ,p is transmitted to the inner ring as a moment about the axis ,e ,f solely because of the friction existing at the journals of the axis ,e ,f4 ,this friction may be very small1 so that the turning moment received by the wheel is only a very small fraction of the turning moment exerted by the weight ,w about ,n ,p4 ,the effect of the arrangement is thus exactly the same as would be produced in the arrangement ,fig4 #a if we reduced the weight ,w to a hundredth or a thousandth of its value4 ,in other words1 precession about the vertical axis ,h ,j will set in in the direction of the arrow ,m just as before1 but the speed of this precession will be only a hundredth or a thousandth of the previous value4 ,it is not very important to trace out the behaviour of the system shown in ,fig4 #c beyond a very brief period immediately after the weight ,w is applied4 ,the point of importance is that the precession produced by the weight is very slow1 and therefore that in a given interval of time the amount precessed is very small4 ,further1 the rate of the precession depends solely upon the friction at the journals of the axis ,e ,f and not upon the weight ,w or the movement of the frame ,k except in so far as these factors affect the friction4 ,the less the friction the less will be the rate of precession and the amount precessed in a given time4 ,thus by mounting the axis ,e ,f on knife edges the friction can be made so small that the precession produced by the weight ,w becomes immeasurable4 ,hence we deduce that if friction is substantially absent at the axis ,e ,f the frame ,k might be violently rocked on the axis ,n ,p or even set into continuous rotation without causing the axle of the wheel either to dip or to precess4 ,continuing the argument1 we might mount the square frame on a vertical axis and attempt to produce rotation of the wheel about the axis ,h ,j by applying a horizontal force to one side of the square frame instead of a force ,v on the outer ring as shown in ,fig4 #a4 ,a similar result would be obtained4 ,granted an all but total absence of friction at the journals of the vertical axis ,h ,j1 the precession produced about the horizontal axis ,e ,f would be immeasurably small4 ,thus the frame might be set into violent motion about its vertical axis without causing the axle either to rotate in a horizontal plane or to precess in a vertical one4 ,illustration3 ,,fig4 #d4 ,one ,degree of ,freedom ,lost4 ,finally1 if the square frame were mounted on a horizontal axis collinear with the axle ,b ,c it might obviously be rotated about this axis without affecting the system otherwise than by increasing or reducing the rubbing speed of the axle ,b ,c in its bearings4 ,since pure translation of the frame in any direction cannot apply a turning moment to the system about any axis1 and as rotation of the frame about any one of the three principal axes has no effect which is measurable on the orientation of the axle1 it follows that1 given substantial absence of friction at the axes ,e ,f and ,h ,j1 the axle of the wheel will remain constantly pointing parallel with its original position1 no matter how the frame ,k may be moved or turned about4 ,illustration3 ,,fig4 #e4 ,second ,degree of ,freedom ,lost4 ,the gyroscopic system shown in ,fig4 #a has1 as we have said1 8three degrees of freedom10 because its wheel is free to spin about three different axes mutually at right angles4 ,it is to be carefully noted that it can only truly be said to have three degrees of freedom so long as the inner ring and the parts inside it are not rotated on the axis ,e ,f away from the position which in ,fig4 #a they are shown as occupying relatively to the outer ring4 ,thus rotation of the wheel on its axle or of the whole system inside the square frame on the axis ,h ,j leaves the three axes ,b ,c1 ,e ,f1 ,h ,j undisturbed at right angles to each other4 ,but rotation of the inner ring and the parts inside it on the axis ,e ,f tends to destroy one of the degrees of freedom4 ,if1 for instance1 the inner ring is rotated through #ij deg41 as shown in ,fig4 #d1 the axle ,b ,c and the axis ,h ,j will coincide in direction4 ,in this position the wheel cannot be rotated about a horizontal axis at right angles to ,e ,f and has therefore virtually only two degrees of freedom1 namely1 about the axis ,e ,f and about the axis ,h ,b ,c ,j4 ,again1 if with the inner ring in the position shown in ,fig4 #d the outer ring is turned through #ij deg4 relatively to the square frame1 the system assumes the configuration shown in ,fig4 #e and the wheel loses the power of rotating about a horizontal axis in the plane of the square frame4 ,illustration3 ,,fig4 #f4 ,lost ,degrees of ,freedom ,restored4 ,if1 then1 in any application of the gyroscope it is necessary to guarantee that the system shall have three degrees of freedom in all possible configurations1 the simple mounting shown in ,fig4 #a will not serve the purpose4 ,it can be made to do so in the manner shown in ,fig4 #f1 namely1 by mounting the square frame inside a gimbal ring ,x1 which in turn is supported by a frame ,y1 the two new axes ,t ,u and ,v ,w being at right angles to each other4 ,in the position shown in ,fig4 #d the new axis ,t ,u would restore the lost third degree of freedom1 while the second new axis ,v ,w would restore the degree of freedom lost when the system assumed the configuration shown in ,fig4 #e4 ,in the gyro-compass it is necessary to guarantee that the spinning wheel in all possible configurations shall have three degrees of freedom1 and accordingly we find the wheel mounted in a manner reproducing the features of ,fig4 #f4 ,on the other hand1 the majority of the movements which the compass system is called upon to make do not entail anything except very small degrees of rotation of the inner ring and wheel about the axis ,e ,f 7,fig4 #a71 and therefore for most purposes the simple mounting there shown reproduces the required three degrees of freedom sufficiently closely to permit us to use it for demonstration purposes4 ,in one very important portion of our subsequent discussion1 however--namely1 that dealing with the effect on a marine gyro-compass produced by rolling and pitching of the vessel--it will be necessary for us to take cognisance of the fact that the square frame shown in ,fig4 #a is not fixed directly to the ship's deck1 but is really carried in gimbals as shown in ,fig4 #f4 ,,chapter ,,iii ,,the ,,gyroscope ,,and ,,the ,,rotation ,,of ,,the ,,earth ,let us now suppose that the gyroscopic system shown in ,fig4 #a--without the weight ,w--is placed on the equator as represented at ,p 7,fig4 #g71 and that the axle is set pointing due east and west as at ,b ,c4 ,at the end of an interval of time1 say two hours1 the earth will have rotated through some angle 1 say #cj deg41 and will have carried the gyroscope with it to the position ,q4 ,the square frame has thus clearly been inclined relatively to its original position4 ,it has1 in fact1 suffered the exact equivalent of a direct translation ,r together with a pure rotation about a horizontal axis through ,e of amount 4 ,the translation leaves the system unaffected1 but the rotation of the frame results in the frame moving relatively to the axle1 wheel1 and inner ring4 ,the axle1 in fact1 remains parallel with its original position at ,p4 ,it is still pointing east and west1 but the frame is now inclined to it and1 relatively to the horizontal surface of the earth at ,q1 the axle is dipping at an angle which is equal to --or #cj deg4 ,actually1 if we fixed a disc to the square frame and a hand to the inner ring1 as indicated in ,fig4 #h1 the system as erected at the equator would form a twenty-four hour clock indicating strictly accurate sidereal time as distinct from mean solar time4 ,to an observer on the earth the hand would appear to travel clockwise round the disc once in twenty-four hours4 ,actually1 however1 the hand would not rotate1 but would remain constantly parallel with its original position1 while the disc would travel anti-clockwise relatively to the hand and would make one complete turn in twenty-four hours4 ,the hand would remain parallel with its original position by virtue of the fact already stated1 namely1 that the force applied to the inner ring through its all-but-frictionless supports is very small1 and in any event does not turn the hand clockwise1 but causes the wheel1 inner ring1 and hand to precess about the vertical axis ,h ,j4 ,the rate at which this precession took place would be a measure of the success with which we had eliminated friction at the horizontal axis ,e ,f4 ,illustration3 ,,fig4 #g4 ,elementary ,gyroscope at ,equator4 ,illustration3 ,,fig4 #h4 ,gyroscopic ,clock4 ,it might be thought that the system would without further addition serve as a compass1 for if it maintains its axle constantly pointing in one direction it is just as good as a compass which always points its needle towards the magnetic north4 ,in the magnetic compass1 however1 the needle has a directive force applied to it which enables it to recover its standard direction if it should be accidentally deflected from it4 ,in the gyroscopic system we have been considering there is no such directive force4 ,the axle will remain pointing in one direction1 it is true1 but the system is indifferent as to what that direction may be4 ,if the axle is accidentally deflected it will remain pointing in the new direction as consistently as it did before in the originally set direction4 ,as a substitute for the ordinary compass1 then1 the success of the device would depend upon the success with which accidental deflecting forces were prevented from acting on the axle after it had been set in a known direction4 ,in practice1 as ,dr4 ,anschtz found in his early investigations1 it is excessively difficult1 if not quite impossible1 to construct a gyroscopic system in which the centre of gravity and the centre of suspension are absolutely coincident4 ,as a result a very minute gravity torque is thrown on the system1 and in consequence the axle very slowly precesses away from the original set direction4 ,this fact and the complication of parts required to give practical effect to the idea led ,dr4 ,anschtz to abandon his early attempts at providing a compass substitute of the apparently simple nature described above4 ,illustration3 ,,fig4 #i4 ,elementary ,gyro-,compass4 ,an addition to the system of a very simple kind in itself not only endows the axle with directive force1 but makes the direction which it seeks the north and south one1 and thus converts the system into a device possessing the familiar property of the compass4 ,this addition consists of a pendulum-like weight ,s 7,fig4 #i71 attached below the wheel by a stirrup fixed to the inner ring so that the weight1 stirrup1 inner ring1 axle1 and wheel may swing as a whole on the horizontal axis ,e ,f4 ,let us suppose that the system with this addition is set up at the equator and that the axle this time is aligned at right angles to the equator so that the end ,b1 as shown at ,i 7,fig4 #aj71 points due north4 ,in this condition the inner ring is horizontal and the weight ,s is vertically below the pendulum axis ,e ,f4 ,no turning moment is therefore being applied by gravity to the wheel4 ,if through imperfection of workmanship the centre of suspension of the system is not absolutely coincident with the centre of gravity before the weight ,s and its stirrup are attached1 then the minute gravity torque arising from the lack of coincidence will be balanced automatically by the weight1 the inner ring taking up some position minutely inclined to the horizontal4 ,there will thus under all conditions be no resultant turning moment applied by gravity to the system as thus set up4 ,in addition1 the axle1 lying north and south as it does1 is aligned parallel with the earth's polar diameter4 ,consequently the rotation of the earth can only move it parallel with its original position1 and therefore does not tend to cause relative motion between it and the square frame4 ,we conclude1 then1 that in this north and south position of the axle the system is not acted upon by any force or influence tending to cause the axle to depart from the north and south position4 ,illustration3 ,,fig4 #aj4 ,elementary ,compass at ,equator4 ,now let us suppose that the axle by some agency is forced into parallelism with the equator so that the end ,b points due east as indicated at ,,ii 7,fig4 #aj74 ,immediately after it takes up this position the tendency of the axle to remain parallel with this1 its new original1 direction becomes manifested in attempted relative motion between the axle1 wheel1 and inner ring on the one hand and the square frame on the other4 ,thus as the earth rotates the axle1 etc41 tend to set themselves relatively to the frame in the position shown at ,,iii4 ,in this position1 however1 the weight ,s being rigidly suspended from the inner ring1 is no longer vertically below the pendulum axis ,e ,f4 ,gravity acting upon the weight therefore applies a turning force to the wheel1 etc41 about the axis ,e ,f4 ,the system is thus under the same conditions as those represented in ,fig4 #a1 when the weight ,w is hung on the inner ring at ,b4 ,precession about the vertical axis therefore sets in1 in the direction ,m 7,fig4 #a71 so that the end ,b of the axle swings round from the east towards the north4 ,let us reset the system in position ,i1 and then by some agency cause the axle again to align itself parallel with the equator1 this time1 however1 with the end ,b pointing due west as shown at ,,iv 7,fig4 #aj74 ,as before1 the rotation of the earth combined with the tendency of the axle to remain parallel with the new west and east position results in attempted relative motion between the axle1 etc41 and the square frame1 so that in a little time the system would adopt the configuration shown at ,v4 ,in this configuration1 however1 gravity as before applies through the weight ,s a turning force about the pendulum axis ,e ,f4 ,now1 comparing the two configurations ,,iii and ,v1 it will be seen that1 mere reference letters or similar distinguishing marks being washed out1 they are indistinguishable except for one fact3 the wheel is rotating in opposite directions4 ,if with the system as arranged in ,fig4 #a we reverse the direction of spin of the wheel without reversing the direction of the applied force ,w1 then1 as we know already1 the direction of the precession will be reversed4 ,precession about the vertical axis will take place in the direction opposed to the arrow ,m4 ,hence in the configuration shown at ,v 7,fig4 #aj71 the precession induced by the action of gravity on the weight ,s causes the end ,b of the axle to swing up from due west towards the north4 ,we are thus led to identify the end ,b of the axle as the north-seeking end and the end ,c as the south-seeking4 ,with ,b pointing due north as at ,i1 there is no force acting on the system tending to make the axle depart from the north and south direction4 ,if ,b is swung over to the east or west--or intermediately1 as may be taken for granted--a force is called into play tending to move the end ,b back towards the north4 ,it follows1 therefore1 that the resting position of the axle is the north and south one with the end ,b pointing north4 ,it may be said1 perhaps1 that we have neglected to discuss what happens if from the position ,i the wheel is turned by some agency right round until the end ,b of the axle points due south4 ,in this condition there is no resultant gravity torque on the pendulum axis1 and the axle is lying parallel with the earth's polar axis1 so that the rotation of the earth does not cause relative motion between the wheel1 etc41 and the frame4 ,just as in the original configuration ,i1 there is thus in this condition no force applied to the system tending to make the end ,b swing away from the pole4 ,but as the reader may readily establish for himself by reversing the arrows and reference letters in the five diagrams of ,fig4 #aj1 the slightest departure of the end ,b of the axle from the south-pointing direction towards either side of the meridian will call into play a force which will cause the end ,b to precess up towards the north4 ,with the wheel spinning in the direction we have shown throughout our illustrations the only stable position of equilibrium for the axle is the north and south with the end ,b pointing north4 ,it may be pointed out that the magnetic needle can1 like the gyro-compass axle1 assume a position of unstable equilibrium with the north-seeking end pointing south4 ,a point of very great practical importance into which to inquire is the magnitude of the directive force1 the existence of which1 when the axle is deflected from the north and south position1 we have just demonstrated4 ,this directive force or restoring moment1 as will have been gathered from our explanation1 increases with the deflection from the north1 being a maximum when the axle is lying east and west or west and east4 ,its magnitude in any position of the axle depends upon 7a7 the speed of rotation of the earth on its polar axis1 7b7 the speed of the spinning wheel on its axle1 and 7c7 the mass1 or1 more correctly1 the moment of inertia1 of the wheel4 ,the first item is small--#j4jjjg of a revolution per minute--and is quite beyond our control4 ,the second factor is consequently made as large as possible1 while the third is also made large1 but a limit is placed to our choice by questions of safety and temperature rise at the high speeds adopted for the spin of the wheel4 ,in the following table we give the values of these factors for three of the types of gyro-compass to be described later4 ,compass ,wheel diam4 ,wheel weight ,speed in4 lbs4 r4p4m4 ,anschtz#a #f #aj #bj1jjj ,sperry #ab #de #h1fjj ,brown #d #d #ae1jjj ,the value of the directive force for the same three gyro-compasses and for an ordinary magnetic compass is given in the next table1 7#a7 for the axle--or needle--lying due east and west1 and 7#b7 for the axle--or needle--deflected #a deg4 east or west of north--true or magnetic4 ,directive ,force at ,equator4 ,axle 7or needle7 ,axle 7or needle7 ,e4 and ,w4 #a deg4 ,e4 or ,w4 of ,n4 ,force ,leverage ,force ,leverage ,grains in4 ,grains in4 ,anschtz #ade #a #b #a ,sperry4 #aadj #a #bj #a ,brown #ab #a #a ,magnetic#b #dj #a #a ,illustration3 ,,fig4 #aa4 ,elementary ,compass at #ee deg4 ,n4 ,lat4 ,we have now to explain how the gyro-pendulum system manifests its compass-like property when it is transferred from the equator to some degree of latitude north or south4 ,in ,fig4 #aa we represent the system as set up in the latitude of the ,british ,isles4 ,the axle is horizontal and the end ,b is pointing due east4 ,in this configuration the earth's rotation is1 through the action of gravity on the pendulum bob ,s1 trying to make the wheel turn round the earth's polar axis once every twenty-four hours4 ,as before1 in accordance with the fundamental property of the gyroscope1 the wheel will try to set its axle into coincidence with or parallel with the axis about which it is being forced to rotate4 ,in other words1 the wheel will endeavour to turn in such a way as to align its axle along ,v ,u with the end ,b towards ,u4 ,this movement can be effected by a rotation about the vertical axis ,h ,j through a right angle combined with a rotation about the horizontal axis ,e ,f through an angle equal to the latitude of the station at which the system is set up4 ,the rotation about the vertical axis ,h ,j does not result in deflecting the weight ,s away from the plumb line1 and therefore can be completely fulfilled4 ,the rotation about the horizontal axis does1 however1 affect the bob4 ,the axle1 having executed the horizontal portion of its movement1 is pointing its end ,b due north1 but this end1 unlike its behaviour at the equator1 manifests a desire to rise vertically so as to align the axle along ,v ,u4 ,its desire to do so is resisted by the bob ,s1 and the axle therefore fails to complete the full movement4 ,the axle is thus held substantially horizontal with its end ,b pointing to the north4 ,as the earth rotates the desire of the axle to align itself parallel with the polar axis persists4 ,in attempting continuously to fulfil this desire it acquires a slight upward tilt1 which is sufficient to bring the pendulum weight into action4 ,with the weight thus slightly deflected towards the north a moment is applied to the wheel which tends to turn the wheel about the horizontal axis ,e ,f in such a way as to bring the end ,b down again to the horizontal plane4 ,such a moment1 as we know from the fundamental rule of the gyroscope1 will actually produce precession about the vertical axis ,h ,j1 the direction of this precession being such as to cause the end ,b to move away from the north towards the west4 ,the fact1 then1 that the axle is prevented from aligning itself completely parallel with the earth's polar axis thus apparently results1 once it has found the north1 in making it wander1 in northern latitudes1 towards the west4 ,this is not so4 ,once the axle has found the north a steady uniform precession towards the west is required to maintain it on the north4 ,thus in ,fig4 #ab let ,a be the wheel and axle of the compass when it finds the north4 ,if the axle maintained this alignment then some time later it would assume the position shown dotted at ,d2 that is to say1 it would be pointing towards east of north4 ,to maintain it on the north it must rotate westwardly through the angle in the interval between ,a and ,d4 ,as this angle grows with the interval the required rotation is really equivalent to a steady uniform precession towards the west4 ,illustration3 ,,fig4 #ab4 ,compass at ,equator and near ,north ,pole4 ,if the compass in ,fig4 #ab is practically at the north pole it is clear that to hold the end ,b of the axle on the north the axle has to precess about the vertical axis ,h ,j of the compass mounting at the rate of practically one complete revolution per twenty-four hours--that is1 #j4jjjg revolution per minute4 ,at the equator the required rate of precession about ,h ,j is zero1 for any movement about this axis will carry the axle away from the north4 ,at intermediate latitudes the precession required to hold the axle on the north has an intermediate value4 ,its value for any latitude is1 in fact1 #j4jjjg sin in revolutions per minute4 ,thus as the latitude is changed the required rate of precession also changes4 ,so1 too1 does the angle 7,fig4 #aa71 by which the axle fails to reach parallelism with the earth's polar axis1 and consequently so does the strength with which the axle desires to reach this alignment4 ,as the equator is left farther and farther behind1 then1 the axle comes to rest pointing north with a greater and greater upward tilt from the horizontal4 ,the applied moment of the weight ,s thus increases4 ,it increases just at the rate necessary to give the required rate of westerly precession for the particular latitude in which the compass is at any moment stationed4 ,should anything prevent the axle from acquiring the tilt appropriate to the latitude1 or should the westerly precession on the vertical axis caused by this tilt be opposed and reduced in any way1 the axle will fail to keep on the north and will lag behind the meridian with an easterly deviation4 ,we shall see later on that the precession about the vertical axis is in some designs of gyro-compass unavoidably opposed1 and that as a consequence these compasses exhibit a latitude error4 ,we have thus shown that the effort of the compass to set its axle parallel with the earth's polar axis1 combined with the action of gravity on the pendulum weight1 is necessary to the compass if the axle once having found the north is to remain on it1 and that this effort of the axle increases in strength the farther north--or south--the compass is moved from the equator4 ,what1 however1 this effort gains in strength as the angle of latitude increases the effective directive force on the compass loses4 ,thus in ,fig4 #aa the directive force may be represented as at f by a line parallel with the earth's polar axis4 ,this line represents the magnitude of the compass's effort to set its axle parallel with the polar axis4 ,the speed of the spinning wheel and its moment of inertia have not been altered by moving the compass away from the equator1 nor has the angular speed of the compass round the earth's axis1 for although the compass is moving in a circle of reduced radius ,t ,b1 and therefore is travelling with less linear velocity than at the equator1 it is still making one turn per twenty-four hours round the polar axis of the earth4 ,thus the three factors fixing the magnitude of the 8directive force0 are unaltered4 ,the force f is thus the same as that exerted on the compass at the equator4 ,it does not1 however1 act as before1 purely about the vertical axis ,h ,j1 but partly about ,h ,j and partly about the horizontal axis ,e ,f4 ,it may be regarded as a force applied at the end ,b of the axle and therefore as tending to turn the wheel about an imaginary axis a b4 ,we may resolve it into two components p and q1 p being at right angles to the axis ,h ,j and q at right angles to ,e ,f4 ,the component q represents the magnitude of the upwardly tilting effect applied to the axle by the rotation of the earth4 ,the component p represents the effective directive force tending to restore the axle from the deflected position represented towards the north in the horizontal plane4 ,the angle between this effective component and the full force f is 1 the angle of latitude of the station at which the compass is set up4 ,the effective directive force is thus f cos 1 and therefore diminishes from the value f at the equator towards zero as the north--or south--pole is approached4 ,,chapter ,,iv ,,damping ,,the ,,vibrations ,,of ,,the ,,gyro-,,compass ,reviewing what we have already established1 we see that a gyroscopic system possessing 8three degrees of freedom0 and having a pendulum weight fixed below the wheel manifests a tendency in all latitudes to preserve its axle pointing in the north and south direction1 a 8directive force0 or restoring moment being developed and applied to the axle if the north and south position is departed from4 ,the magnitude of the directive force in any given latitude increases with the deflection1 from zero when the axle is pointing north and south up to a maximum when it is aligned east and west4 ,at any given angle of deflection of the axle the magnitude of the directive force varies with the latitude in which the system is stationed1 being zero at the north or south 7true7 pole and a maximum at the equator4 ,finally1 at any given angle of deflection of the axle and in any given latitude the magnitude of the directive force is determined by 7a7 the speed of rotation of the earth on its polar axis1 7b7 the speed of rotation of the spinning wheel on its axle1 and 7c7 the mass or moment of inertia of the spinning wheel4 ,we have now to consider several important matters affecting the practical application of the gyroscope-pendulum combination as a substitute for the magnetic compass4 ,the first practical consideration which arises naturally in our minds is the question3 ,can a system be devised and constructed sufficiently robust to withstand the trials and knocks of every-day use and yet be sufficiently delicate to respond to the feeble directive forces on the effect of which its action as a compass depends8 ,from the table given previously it will have been noted that in the three chief types of gyro-compass so far developed the directive forces developed are in two examples greater than the corresponding directive force applied to the card of a magnetic compass1 while in the third the directive force is materially lower4 ,even though they were all considerably greater than the force applied to the needle of a magnetic compass1 some doubt as to their sufficiency to effect their work would remain1 for they have to control sensitive elements1 comprising a spinning wheel1 axle1 supporting rings1 etc41 weighing anything from #g lb4 to about a hundredweight1 whereas in the ordinary compass the sensitive element consisting of the card and its attached magnetic needles weighs round about oz4 ,the actual weights of the sensitive elements are given in the following table4 ,weight of ,sensitive ,element4 ,anschtz compass #ae lb4 ,sperry compass #ajj lb4 ,brown compass #g lb4 ,thomson magnetic compass #agh grains ,whether or not the directive force developed will be sufficient to control the movement of the sensitive element in a gyro-compass must clearly depend very largely upon the degree of success reached in banishing friction from the vertical axis about which the sensitive element moves4 ,without for the present describing the means actually adopted to secure virtually frictionless support in the three types of gyro-compass1 we may say that were they other than very refined no compass action whatever would be manifested4 ,in the early theoretical days of the gyro-compass before sufficiently refined practical constructive methods had been developed1 the experimental verification of the mathematical results arrived at could not be attempted4 ,granted the attainment in practice of a satisfactory frictionless method of supporting the sensitive element1 we have next to note that the simple gyro-pendulum system which we have been considering would be quite useless as a direction-finding device either at sea or on land by virtue of the fact that the very absence of friction at the vertical axis would encourage the sensitive element to oscillate from side to side of the meridian under the least provocation4 ,the period of oscillation would be a prolonged one1 much too prolonged1 in fact1 to permit the true north to be determined by taking the mean of the extreme positions reached by the gyro-axle in the course of its oscillation4 ,we have seen that the simple gyro-pendulum system which we have so far been considering1 when placed on the equator1 manifests a tendency to set its axle north and south1 that if the axle is deflected towards the east a westerly turning directive force is developed1 and that if the axle is deflected towards the west an easterly turning directive force is developed4 ,in an ordinary vertical pendulum 7,fig4 #ac71 the resting position of the bob is at d4 ,if it is swung to the position e--towards the east1 let us say--the weight w of the bob supplies a moment about the axis at g1 tending to restore the pendulum to its resting position2 while if it is swung towards the position f--towards the west1 we may suppose--the moment is reversed and again acts to restore the pendulum to its resting position4 ,illustration3 ,,fig4 #ac4 ,pendulum and ,compass4 ,the gyro-pendulum system as set up at the equator is1 it will be seen1 subjected to an exactly analogous set of forces when its axle is deflected east or west4 ,the system1 in fact1 constitutes virtually a horizontal pendulum1 the vertical axis ,h ,j being identified with the axis at g in the ordinary pendulum4 ,now we know that if we deflect an ordinary pendulum to some such position as e and let it go it will swing through the resting position d to a position f equidistant on the other side1 and will continue to vibrate until friction at the axis g1 air resistance1 etc41 sap its original stock of energy communicated to it by the initial deflection4 ,the period of vibration--the time elapsing between two successive passages of the bob in the same direction through the resting position--is determined by the length of the pendulum and remains constant throughout1 even when the amplitude of the swing has fallen off virtually to nothing4 ,an exactly analogous state of affairs exists with the gyro-pendulum system4 ,if the axle is deflected towards the east and then let go it will swing back under the action of the directive or restoring force through the north and south position over to an equal angle on the western side1 and will thereafter vibrate back and forth with a constant period1 until frictional and other losses cause the motion to die away4 ,the period of vibration is determined by a complication of factors1 among which are the speed at which the wheel is spinning on its axle1 the speed of rotation of the earth1 and the mass of the sensitive element4 ,if the sensitive element can be regarded as consisting solely of the wheel1 then1 no matter what may be the size of the wheel1 so long as it is in the form of a circular disc1 the period of vibration is determined solely by the speed of the wheel and the speed of rotation of the earth4 ,for a wheel at the equator running at #bj1jjj revolutions per minute the period of vibration would be about eleven seconds4 ,in practice1 however1 the weight of the axle1 the inner and outer supporting rings--or their equivalents--the pendulum bob and various other fittings and adjuncts of a secondary nature have to be added to the weight of the wheel in assessing the influence of the sensitive element upon the period of vibration4 ,the greater the mass--or more correctly1 the moment of inertia--of the sensitive element the longer will be the period of vibration4 ,in the early--#aiaj--;,anschtz gyro-compass the sensitive element had a moment of inertia such that the period of vibration at the equator was just over #fa minutes2 that is to say1 #ccd times as long as it would have been if the sensitive element had consisted of nothing but the spinning wheel4 ,this very prolonged period1 were nothing done to rectify matters1 would be a very serious objection in practice to the use of the gyro-compass4 ,the axle1 if deflected1 would take about half an hour to reach an equal position on the opposite side of the meridian4 ,hence1 if1 when a compass reading was desired1 the axle were found to be vibrating1 at least half an hour would be required to determine the north and south direction by observing the two extreme positions of the axle and taking the mean4 ,the alternative would be to wait until the vibration died away4 ,this course would involve1 however1 a very much greater delay1 for the virtual absence of friction at the vertical axis of the system--an essential1 as we have seen1 if the directive force is to be allowed to come into play at all--results in the vibration being practically unchecked1 so that1 once started1 it would continue almost indefinitely4 ,some means of damping the vibration analogous to the damping action of the liquid in a magnetic compass must clearly then be provided4 ,ideally the means should be such that if the axle is deflected through any angle it will return to the north and south position in a 8dead-beat0 manner and not swing across the meridian over to the opposite side4 ,this ideal cannot be realised in practice4 ,returning to the simple pendulum illustrated in ,fig4 #ac we have to notice that the influence at work causing the vibration is the weight of the bob acting about the axis at g4 ,this influence is a maximum when the bob is at the extreme positions e f and is zero when the bob is at d4 ,on the other hand1 the velocity of the bob is zero at the two extreme positions e f and is a maximum at d4 ,during the swing from e to d the vibrating influence is helping the motion of the bob and the velocity consequently increases4 ,at d the vibrating influence disappears1 while during the swing from d to f it reappears and this time opposes the motion of the bob1 the velocity of which consequently becomes less and less4 ,the movement of the bob from d to f in opposition to the vibrating influence is achieved by the momentum of the bob arising from the velocity which it gathers during its swing from e to d4 ,for the angle d g f to be equal to the angle d g e the velocity of the bob as it passes through the position d must just be a certain amount1 no more and no less1 namely1 the velocity which a body would acquire in falling from rest at the level of the bob at e vertically downwards to the level of the bob at d4 ,if the velocity of the bob when it swings through d is greater than this amount the bob will swing beyond the position f4 ,if it is less the bob will fail to reach f4 ,illustration3 ,,fig4 #ad4 ,damped and ,undamped ,vibrations4 ,the analogue of the problem to be solved in connection with the gyro-compass is to devise some means that will rob the pendulum bob on each successive swing of some percentage of the velocity with which it passed through the resting position d during the preceding swing4 ,by so doing we shall obviously decrease continuously the angle to which the pendulum swings on each side of the position d4 ,thus instead of the swings as traced out on a piece of paper moving below the bob being as shown at ,a 7,fig4 #ad71 they will be of the form represented at ,b4 ,the amplitudes1 instead of remaining of uniform amount practically indefinitely1 will diminish with each swing until they become so small as to be invisible4 ,it is to be noted that theoretically the vibrations cannot be completely suppressed even after an indefinite number of swings1 for if the velocity at the resting position is at each swing1 say1 #ej per cent4 less than at the previous passage1 it will always be something and never become zero4 ,it will1 however1 in quite a small number of swings become so low that the motion of the pendulum will be practically undiscernible4 ,thus with a #ej per cent4 decrement the velocity at the eighth passage of the bob through the resting position will be less than #a per cent4 of what it was at the first passage4 ,it is also to be noted that while the amplitudes are decreased in the manner indicated the periods of the swings are not being made less4 ,in an ordinary pendulum the period1 as we have said1 depends solely upon the length and--within quite wide limits at least--remains the same whatever be the angle to which we originally deflect the bob4 ,we should therefore expect that if the swings are 8damped0 in the way shown at ,b 7,fig4 #ad71 the period of each swing would be the same and equal to that of the undamped swings represented at ,a4 ,actually the period of a damped vibration is always somewhat greater than that of the same system vibrating freely1 for by robbing the pendulum of some of its velocity at each swing we are virtually causing the bob to pass through the resting position with the velocity of a free swinging pendulum of greater length and therefore of increased period4 ,the increase in the period of the damped pendulum over the same pendulum when undamped is determined by the strength of the damping means employed1 or1 in other words1 by the percentage by which we reduce the velocity at each swing4 ,in the early 7#aiaj7 ,anschtz compass the period of vibration at the equator without damping was1 as we have stated already1 about #fa minutes4 ,with its damping device in action the period of the compass at the equator became approximately #gj minutes4 ,in later designs of gyro-compasses the period of the damped vibration is deliberately made #he minutes or thereabouts4 ,a practical advantage--to be explained later--is secured by adopting this particular value4 ,it is the period which a simple pendulum would have if its length were equal to the radius of the earth--#djjj miles or so4 ,illustration3 ,,fig4 #ae4 ,damped ,pendulum4 ,a vibrating pendulum 7,fig4 #ae7 will be satisfactorily damped if we can apply to the bob in all positions of its swing a force proportional to the velocity with which the bob is moving at each given instant and directed always so as to oppose the motion4 ,at e1 the position of release after deflection1 the bob has no velocity2 the damping force should therefore be zero4 ,as it travels from e to d the bob is gaining velocity2 the damping force should therefore grow from zero to a maximum at d and be directed at each instant tangentially to the curve of swing and act from d towards e4 ,during this portion of its swing the bob is thus robbed of some of its velocity1 so that it fails to rise to the position f and comes to rest at some point g4 ,in travelling from d to g the velocity of the bob is decreasing naturally1 and is still further decreased by the damping force which1 acting in the same direction as before1 falls from a maximum at d to zero at g4 ,as it moves from g to d on the return stroke the bob gains velocity2 the damping force should therefore increase from zero at g to a maximum at d--of a lower value than the maximum at the same point during the first stroke--and should be directed along the curve in the reversed sense1 namely1 from d towards g4 ,during the swing from d to h the damping force1 still reversed1 should decrease from the second maximum value once more to zero4 ,at the start of the second swing the damping force should again act from d towards e and should rise from zero to a third maximum value at d--less than either the second or first maximum value1 and so on until the amplitude of the swing is reduced to the required degree4 ,it will be noticed that in a damped vibration the mean position of the bob on any one swing is not coincident with the resting position d1 but lies somewhere between the resting position and the position from which the bob commences the swing4 ,with each passage of the bob through the resting position d the value of the damping force rises to a maximum1 the value of which becomes less and less on each successive swing4 ,ultimately1 when the bob settles in the resting position1 the maximum value becomes zero4 ,in other words1 the damping force1 having completed its work by bringing the bob to rest1 entirely disappears and leaves the pendulum exactly in the same condition as it would be under in the resting position if no damping force had ever been in action4 ,the pendulum is thus as free as formerly to respond1 in the resting position1 to a vibrating influence1 but as soon as it acquires motion the damping force is again called into existence to a degree directly dependent on the strength of the vibrating influence1 with the result that the motion is first checked1 and then finally stopped4 ,the damping force required is1 as we have said1 one which at all times is proportional in magnitude to the velocity of the bob--or what is the same thing1 to the angular velocity of the pendulum as a whole--and which at all times acts to oppose the motion of the bob4 ,metallic friction--say1 at the supporting axis of the pendulum--it would bring the motion to rest sooner or later1 would not provide a satisfactory damping force1 for solid friction is independent of the rubbing velocity1 at least at low speeds such as we are here concerned with4 ,the damping force provided by it being constant1 would not be automatically adjusted to the velocity of the bob4 ,it would vanish1 it is true1 when the bob was at rest1 but as soon as the slightest vibration set in it would spring up to its full value straight away and would preserve the same value throughout a large swing as throughout a small one4 ,in any event the presence of metallic or other solid friction at the point in the gyro-compass corresponding to the axis of the pendulum--namely1 at the bearings of the vertical axis ,h ,j--cannot be permitted1 and must be eliminated to the utmost possible degree if the directive force is to be sufficient to control the movement of the sensitive element4 ,fluid friction1 on the other hand1 would provide a satisfactory damping force1 for fluid friction is proportional to the velocity1 at least at low speeds4 ,a pendulum vibrating with its bob in a vessel of water or the floating card of an ordinary magnetic compass is satisfactorily damped by fluid friction4 ,in the gyro-compass1 however1 the motion to be damped is1 as we have seen1 an exceedingly slow one1 slower in fact than the small hand of a watch if the deflection of the axle from the meridian is initially less than #aa deg4 east or west4 ,a fluid damping force would be proportionately low1 so that without making the damping elements of enormous size the force derived would be insignificant and next to useless for practical purposes4 ,as an illustration of this statement it may be remarked that in the early ,anschtz compass the sensitive element was virtually floated in a bowl of mercury4 ,yet the drag of the mercury1 the velocity of the vibration being so small1 did not measurably reduce the amplitudes of the vibration during observation extending over several hours4 ,this example is not quite a good one1 however1 for the friction at the surface of a body immersed in mercury would appear to be not of the fluid description1 but of the solid type4 ,solid and fluid friction being thus ruled out1 at least as direct means of providing the required damping force1 we have to find some other method of applying it4 ,it is1 or should be1 clear that in whatever way the damping force is applied it should originate within the sensitive element itself4 ,if it originates outside1 then its transmission to the sensitive element cannot1 in view of the fact that its origination1 growth1 and decay are to be controlled by the motion of the element1 be effected in any conceivable way without the introduction of some material connection between the element and the outside source of the force4 ,such a connection can only be made frictionless if the outside source moves in exact unison with the sensitive element4 ,if it does so move it clearly ceases to be an outside source and becomes really part of the sensitive element itself4 ,this consideration suggests generating and applying the damping force gyroscopically by the exertion of some suitable action on the spinning wheel itself4 ,,chapter ,v ,,the ,,damping ,,system ,,of ,,the ,,ansch,,tz 7#aiaj7 ,,compass ,in the preceding chapter we demonstrated the necessity for damping the horizontal oscillatory movement of the gyro-compass axle and discussed the nature of damped vibrations in general4 ,we now turn to describe the damping means provided in each of the practical forms of gyro-compass so far evolved4 ,turning to the early 7#aiaj7 form of ,anschtz compass1 we find that1 as shown in the purely diagrammatic sketch given in ,fig4 #af1 the spinning wheel is enclosed within a metallic case formed with tubular journals ,b ,c for the axle1 and provided with trunnions ,e ,f1 whereby it is supported within the vertical ring1 which1 as before1 is free to turn on a vertical axis ,h ,j4 ,the casing1 it will be seen is1 so far as the support of the wheel is concerned1 exactly equivalent to the horizontal inner ring of our preceding illustrations4 ,in the actual instrument the method of supporting the casing so as to permit it to turn about a horizontal and a vertical axis is not in the least like that shown in our sketch4 ,the wheel1 running at #bj1jjj revolutions per minute1 although it is quite plain1 has a powerful blower-like action4 ,on one side of the casing an orifice ,d is formed for the inlet of air1 and on the periphery below an outlet duct ,k1 directing the air blast tangentially away from the casing1 is provided4 ,the exact value of the pressure of the blast in the early ,anschtz compass is not known to us1 but in the ,brown compass1 wherein a similar blast is developed1 the wheel1 running at #ae1jjj revolutions per minute1 gives an air pressure equal to some #c in4 of water4 ,illustration3 ,,fig4 #af4 ,air ,blast ,damping ,system of ,anschtz 7#aiaj7 ,compass4 ,the mouth of the outlet duct ,k is partially closed by a plate ,l fixed at the end of a pendulum arm suspended frictionlessly1 or practically so1 from some convenient point on the casing1 so that when the casing turns on the vertical axis ,h ,j the arm and plate turn with it4 ,the pendulum arm is carefully balanced in such a way that when the axle of the spinning wheel is horizontal the plate ,l exactly divides the orifice ,k1 leaving equal passages for the air blast on each side4 ,in this condition the two passages ,m ,n being equal in area1 the air blast is divided by the plate into two streams of equal volume and momentum1 so that if their free discharge is not influenced by surrounding objects their reactions on the casing1 one on the one side of the vertical axis ,h ,j1 the other on the opposite side1 will be equal4 ,let the compass be deflected until its axle points east and west1 the end ,b being towards the east4 ,then1 as we have seen1 the tendency of the axle to remain parallel with its original position1 combined with the rotation of the earth1 will cause the axle to assume an inclined position relatively to the earth's horizontal surface4 ,gravity acting on the pendulous weight ,s as thus displaced from the plumb line will1 as we know1 set up a precession about the vertical axis ,h ,j1 so as to cause the end ,b of the axle to move towards the north4 ,when1 however1 the axle tilts in this manner about the horizontal axis ,e ,f1 the pendulum plate ,l1 hanging freely1 remains in the plumb line4 ,consequently the equal areas ,m ,n become unequal1 as at ,p ,q1 the larger ,p being towards that end of the axle which has tilted upwards1 namely1 the north-seeking end ,b4 ,the reaction on the casing of the portion of the air blast issuing from ,p is now greater than that of the portion emitted through ,q4 ,this inequality results in the application to the sensitive element of a force acting about the vertical axis ,h ,j4 ,the reaction of a jet of air or water or other fluid being opposed to the direction in which the jet is issuing1 the force applied to the casing is such as to drive ,p into the plane of the paper and to bring ,q out of it--that is to say1 to tend to rotate the casing on the vertical axis in the direction of the arrow ,r4 ,the force thus applied clearly tends to oppose the return of the axle from the east towards the north4 ,on the other hand1 if the axle succeeds in turning round until the end ,b points to the west1 the rotation of the earth will result in the axle tilting as before1 but with the end ,b downwards4 ,consequently the area ,m diminishes and the area ,n increases1 and the reaction applied to the casing is such as to tend to turn it in the reverse direction to ,r4 ,the applied force in this case therefore opposes the return of the axle from the west towards the north4 ,the exact manner in which the opposition acts is to be noted4 ,a force applied to a simple gyroscope1 so as to tend to make it turn about the vertical axis ,h ,j in the direction ,r 7,fig4 #af7 will1 as was stated in our second chapter1 cause the wheel to precess about the horizontal axis ,e ,f1 the end ,b of the axle going down and the end ,c up4 ,if the force about ,h ,j is reversed1 the movement produced will be a precession of the wheel on ,e ,f such that the end ,b of the axle rises and the end ,c falls4 ,the opposition between the moment applied to the sensitive element by the air blast and the moment applied to it by the directive force is therefore not quite direct4 ,when the end ,b of the axle is deflected towards the east a directive force is called into being by virtue of the tendency of this end to rise4 ,the air blast reaction about ,h ,j induces precession on the axis ,e ,f1 causing the end ,b to fall4 ,by the amount by which the air blast reaction succeeds in lowering the end ,b of the axle1 by that amount will it reduce the normal magnitude of the directive force4 ,when the axle swings through the meridian towards the west a reversed directive force is called into play by virtue of the tendency of the end ,b to fall4 ,the opposition of the air blast reaction arises from the fact that its tendency now is to make the end ,b rise4 ,thus on both sides of the meridian the opposition of the air blast reaction is effective because it tends to precess the wheel on the axis ,e ,f in the direction opposed to that in which the rotation of the earth is trying to tilt the axle4 ,the tilt moves the pendulous weight ,s away from the vertical1 so producing the directive force2 the air blast reaction reduces the tilt and so opposes the directive force4 ,the reaction applied to the sensitive element by the air blast thus fulfils one requirement of a satisfactory damping force2 its effect at all times is opposed in direction to the direction in which the axle is moving4 ,the second requirement is that the magnitude of its effect should always be proportional to the velocity with which the axle is moving4 ,in connection with this second requirement it is assumed that the momenta of the air jets issuing from the openings ,p ,q 7,fig4 #af7 are at all angles of tilt equivalent to the momentum of a single jet issuing from an imaginary orifice ,u situated at some distance d from the axis ,h ,j on the side of the larger opening ,p1 the area of the imaginary orifice ,r and the velocity of the air through it being constant at all angles of tilt4 ,if this assumption is correct1 then the moment about ,h ,j applied by the reaction of the air blast is proportional to the distance d--that is to say1 to the angle of the tilt4 ,the assumption here made is1 we believe1 substantially justified if the angle of tilt is never very great4 ,in actual practice it is always small4 ,various considerations1 however1 suggest that the reactions of the two jets ,p ,q are not equivalent strictly to the reaction of a single jet through an orifice ,r of constant area4 ,thus from geometrical considerations we can show that the sum of the areas ,p ,q is not equal to the sum of the areas ,m ,n4 ,again1 the total weight of air drawn in per minute through the orifice ,d may be constant1 and therefore the total weight of air delivered per minute through the combined openings ,p ,q may be unaffected by the tilt4 ,but the ratio in which the total volume divides itself between the two openings ,p ,q and the velocity through each certainly vary with the tilt4 ,a peculiar practical phenomenon also has to be considered in this connection4 ,in the #aiaj form of ,anschtz compass the peripheral speed of the spinning wheel was #ejj ft4 per second1 or #cdj miles an hour4 ,the air friction at this speed was so very great that after the wheel had been run a few thousand hours its surface was found to be noticeably smoother than it was when the wheel left the grinding machine on which it was finished4 ,as a result of this polishing effect1 we should expect that even though the speed of the wheel remained perfectly constant1 its blower-like action would decrease somewhat until the compass had been in use for a certain length of time4 ,if the blower action does so decrease the magnitude of the air blast reaction on the sensitive element at any given angle of tilt must diminish with time4 ,we do not know whether the diminution would be sufficiently great to introduce a serious error in the reading of the compass4 ,taking the assumption to be correct1 at least for small angles of tilt1 we have next to study how the angle of tilt varies as the axle swings from the east side of the meridian over to the west and back again4 ,when the axle is pointed due east the rotation of the earth1 as we have seen1 tends to make the end ,b rise4 ,if it is pointed due west1 the end ,b tends to fall4 ,if it is lying dead on the meridian1 the earth's rotation has no tilting effect at all on the axle4 ,if the axle is pointed in some direction between due east and due north1 the tilting effect is less than it would be if it were pointed due east1 but it is in the same direction2 the end ,b tends to rise4 ,similarly1 in any position in the north-west quadrant the end ,b tends to tilt downwards under a tilting influence which is somewhat less than that experienced in the due west position4 ,let1 then1 the axle be turned to point due east4 ,the end ,b begins at once to rise1 but immediately the axle thus leaves the horizontal position it begins to feel the directive force1 and it commences to turn towards the north4 ,until it reaches the north the tilting influence of the earth's rotation continues so that during the whole time the axle is swinging through the north-east quadrant the tilt is increasing and the end ,b is rising1 higher and higher4 ,when the north direction is reached the tilting influence vanishes and the end ,b of the axle tends to travel horizontally in its elevated position over into the north-west quadrant under the influence of the momentum acquired in that direction during its movement through the north-east quadrant4 ,but on passing to the west side of the meridian it again comes under the tilting influence of the earth's rotation1 which this time tends to make the end ,b fall4 ,when the axle reaches the due west position1 the tilting influence being equal to that acting over the north-east quadrant and reversed in direction1 and having acted for the same length of time1 will just have wiped out the elevation of the end ,b1 and the axle will again be horizontal4 ,on the return journey through the north-west quadrant the tilting influence is still reversed1 so that on reaching the north position again the end ,b is as far below the horizontal as it was above it on the swing from east to west4 ,passing to the east side of the meridian1 the tilting action assumes its former direction1 the end ,b begins to rise1 and when the due east position is reached again the axle is once more horizontal4 ,the path traced out by the end ,b during a complete vibration from east to west and back again is thus in the nature of an ellipse--or1 in extreme cases1 a circle--such as that shown at a b c d e in ,fig4 #ag4 ,illustration3 ,,fig4 #ag4 ,free and ,damped ,motion of ,axle4 ,on an east-to-west swing1 then1 the end ,b of the axle in ,fig4 #af1 if the air blast is out of action1 tilts upwards from zero to a maximum in the mid position and then again falls to zero4 ,if the air blast is brought into play1 the reaction1 being1 as we have seen--at least for all small angles of tilt--proportional to the angle of tilt of the axle1 will correspondingly increase from zero to a maximum and again fall to zero4 ,on the west-to-east swing the reaction will similarly rise and fall in the reversed direction4 ,the unrestrained movement of the sensitive element under the influence of the directive force is1 as we know1 analogous to that of a simple pendulum1 so that the velocity with which the axle moves is zero in the extreme positions and a maximum in the mid position4 ,thus we see that the air blast reaction not only constantly opposes the movement of the axle1 with the variation in the velocity with which the axle moves4 ,the air blast reaction therefore completely fulfils the requirements of a satisfactory damping force4 ,when it is brought into play the end ,b of the axle1 instead of vibrating indefinitely in a closed elliptical path a b 444 e 7,fig4 #ag7 moves spirally1 as indicated at f g h 4441 until in a relatively short period of time it comes virtually to rest at m pointing towards the north4 ,illustration3 ,,fig4 #ah4 ,damping ,curve from ,anschtz 7#aiaj7 ,compass4 ,an actual curve taken from an ,anschtz 7#aiaj7 compass while it was settling down on the meridian after the gyro-axle had been deflected nearly #de deg4 to one side is given in ,fig4 #ah4 ,the crests ,a ,b ,c occur at #gj-minute intervals--the period of vibration of the system when damped1 as we have already stated4 ,the oscillations are1 it will be seen1 damped down very effectively1 being entirely eliminated in less than three hours in the course of the third complete vibration4 ,it follows that if the wheel of a gyro-compass is started spinning with the axle pointing elsewhere than due north1 several hours must be allowed to elapse before readings are taken from the card4 ,during the period of settling down1 and especially during the later portions1 the movement of the axle towards its resting position is extremely slow1 and cannot be detected by direct observation4 ,it can1 however1 be inferred from the readings of a spirit level placed on the card1 for1 as we have seen1 the oscillations are accompanied by a tilting of the wheel case on its horizontal axis4 ,,chapter ,,vi ,,the ,,damping ,,system ,,of ,,the ,,sperry ,,compass ,in the ,sperry gyro-compass the damping system adopted is mechanically of a very different nature from that used in the early ,anschtz1 although the theoretical principle of action in both cases is the same4 ,the ,sperry method rules out the employment of air or other fluid in any shape or form as a means of generating1 applying1 or transmitting the damping force1 the reason being that if air or other fluid is relied upon1 the damping force--or so the ,sperry ,company holds--will not act in strict unison with the oscillations1 but will invariably lag behind4 ,the details of the ,sperry method are indicated in a diagrammatic manner in ,fig4 #ai4 ,as in the ,anschtz compass1 the spinning wheel revolves in a casing which1 being provided with trunnions ,e ,f1 takes the place of the inner horizontal supporting ring of our elementary gyroscope4 ,since no blowing action is required of the wheel in this compass the casing1 in order to reduce the expenditure of power required to drive the wheel1 is exhausted of air until a vacuum of not less than #bf in4 is registered on a gauge which forms a permanent fixture on the casing4 ,the exhaustion is effected by attaching a hand-operated vacuum pump to a nipple on the casing4 ,the vacuum produced at one exhaustion remains effective for at least a month under proper treatment4 ,that it is very well worth while exhausting the casing1 if the general design of the compass permits it1 is shown by the fact that in the #aiaj ,anschtz compass over #ie per cent4 of the work done by the motor driving the spinning wheel was spent against windage and air friction4 ,illustration3 ,,fig4 #ai4 ,damping ,system of ,sperry ,compass4 ,the outer ring ,g 7,fig4 #ai7 within which the casing is carried is1 as before1 mounted on a vertical axis ,h ,j4 ,a second outer ring ,k--or 8phantom ring10 as it is called--surrounds the ring ,g1 and is mounted co-axially with it4 ,while the ring ,g1 as before1 moves along with the wheel and its casing relatively to the square frame under the influence of the directive force1 the ring ,k is caused to follow it up in exact agreement by means of a small electric motor1 the pinion of which engages with a gear wheel ,l on the upper trunnion1 the current to the motor being automatically controlled by the movement of the ring ,g4 ,the compass card may be regarded as attached directly to the top face of the gear wheel4 ,a second pinion gearing with the wheel ,l can be arranged to transmit the reading of the card to any number of repeater compasses stationed elsewhere4 ,in all our preceding illustrations we have shown the pendulous weight ,s as being attached by a stirrup directly on to the inner horizontal ring or its equivalent1 the wheel casing1 so as to move in rigid connection therewith4 ,in the early ,anschtz compass the design definitely reproduces this arrangement1 but in the ,sperry compass matters are otherwise4 ,the pendulous weight ,s 7,fig4 #ai7 is carried on a stirrup1 which is forked at each end so as to span the rings ,g ,k1 and which is free to swing on pivots ,m ,n fixed on the phantom ring ,k4 ,the pivots ,m ,n are exactly in line with the pivots ,e ,f1 and as the phantom ring ,k and the ring ,g always move in unison1 the two sets of pivots remain at all times collinear4 ,so far the arrangement of parts is exactly similar to that which would be obtained in our simple gyro-pendulum system if the stirrup of the pendulous weight were not fixed rigidly to the inner horizontal ring1 but were swung freely on the pivots ,e ,f4 ,it can be brought into complete identity with the arrangement of our simple system if the pendulous weight ,s 7,fig4 #ai7 or 8bail10 as it is called by the makers1 be provided with a pin at its mid point to engage with a hole in the periphery of the casing4 ,the system1 as thus arranged1 would merely be a distinctly complicated mechanical variation of our simple gyro-pendulum arrangement1 and1 as a compass1 would be open to the same practical objection1 namely1 the persistence with which any oscillation of the axle1 once set up1 would continue4 ,the vibrations would1 in fact1 be quite undamped4 ,the generation and application of a satisfactory damping force is accomplished in a very simple1 yet beautiful and really ingenious1 manner by displacing the pin connecting the bail and the casing from the mid position to some position lying eastwards of the vertical axis ,h ,j1 as shown at ,q4 ,illustration3 ,,fig4 #bj4 ,action of ,excentric ,pin in ,sperry ,compass4 ,in order to follow the action of this arrangement1 let us consider a disc 7,fig4 #bj7 swung on a horizontal axis ,e ,f on which there is also swung a weighted stirrup ,s1 the stirrup and disc being connected by a pin ,q4 ,let this system of parts be held horizontal1 and in the first instance let the pin1 as shown in the upper view1 be arranged on the centre line of the disc4 ,then the weight ,w of the stirrup applies to the disc three forces1 namely1 a force ,w w acting downwards at the end of the pin1 and two upward forces ,p equal to each other at the pivots ,e ,f1 as will readily be seen by considering the stirrup as a lever fulcrumed at the disc end of the pin4 ,the only force tending to turn the disc is the force ,w w acting about the axis ,e ,f4 ,let1 now1 the pin be situated excentrically relatively to the axis ,h ,j of the disc1 as shown in the lower view4 ,the forces applied to the disc by the weight ,w of the stirrup are again three in number4 ,the force applied at the end of the pin is1 as before1 ,w w1 but at the pivot ,e the force applied falls to ,p - p1 while the force at the pivot ,f rises to ,p p4 ,these forces apply turning moments to the disc4 ,about the axis ,e ,f the turning moment1 as before1 is that of the force ,w w4 ,this force1 owing to the displacement of the pin1 has now1 in addition1 a turning moment about the axis ,h ,j in the direction of the arrow ,r4 ,the upward force ,p - p on the pivot ,e tends to turn the disc in the same direction1 while the force ,p p on the pivot ,f tends to turn it in the opposed direction4 ,these three turning moments about the axis ,h ,j exactly balance each other1 as can be confirmed by working out their values4 ,thus there is no net alteration produced by shifting the pin from its central position1 for in all positions of the pin the only effective moment applied by the weight ,w to the disc is that of the force ,w w about the axis ,e ,f4 ,now1 in the ,sperry system the forces ,p - p and ,p p are not allowed to act on the trunnions ,e ,f of the wheel casing1 but are borne by the phantom ring ,k 7,fig4 #ai7 whence they are transmitted back to form part of the load on the follow-up motor4 ,acting on the casing1 therefore1 there remains only the force ,w w4 ,this force exerts a turning moment about the axis ,e ,f strictly equal to that produced by a directly attached pendulous weight4 ,in addition1 it exerts a turning moment about the axis ,h ,j4 ,illustration3 ,,fig4 #ba4 ,action of ,excentric ,pin in ,sperry ,compass4 ,as actually constructed1 the wheel casing and bail in the ,sperry compass cannot be placed in the position shown in ,fig4 #bj1 for the forks at the end of the bail permit the bail1 and therefore the casing1 to swing only through a small angle from the vertical position4 ,it is clear1 however1 that if the parts could be moved into the horizontal position the moment of the force ,w w about the axis ,h ,j would be a maximum1 just as in this position the moment of the same force about the axis ,e ,f is a maximum4 ,it is also clear that when the parts are in the vertical position the force ,w w has no turning moment about either axis4 ,in an intermediate position 7,fig4 #ba7 the moment of the force ,w w about the axis ,e ,f is proportional to a c--that is1 to the component of a b perpendicular to the plane of the casing or disc--just as it would be if the pin were central or if the bail formed a rigid part of the casing4 ,the moment of ,w w about the axis ,h ,j is likewise proportional to the component a c4 ,as the value of the component a c1 at least for small angles of swing1 is proportional to the angle of swing 1 the net result of making the pin excentric and introducing a phantom ring is to apply to the casing when it is deflected 7#a7 the ordinary moment of the weight ,w about the axis ,e ,f1 and 7#b7 an additional moment about the axis ,h ,j4 ,this latter moment is proportional to the angle of swing 1 and in the position shown in ,fig4 #ba tends to turn the casing in the direction of the arrow ,r4 ,if the deflection is towards the other side of the axis ,e ,f1 the moment applied about ,h ,j will clearly tend to rotate the casing in the opposite direction1 as shown at ,t4 ,comparing ,figs4 #ba and #af1 it will be seen that the excentricity of the pin in the ,sperry compass secures exactly the same result as the air blast reaction produces in the ,anschtz compass4 ,it is to be noted that the excentric pin in the ,sperry compass is displaced towards the east when the axle is resting on the meridian4 ,if it were displaced towards the west the force applied about the vertical axis would be reversed in its effect1 and would tend to increase the oscillations of the sensitive element and not to damp them1 as is the intention4 ,,chapter ,,vii ,,the ,,damping ,,system ,,of ,,the ,,brown ,,compass ,in both the ,sperry and the early ,anschtz compass the natural oscillation of the sensitive element about the vertical axis is damped by applying a retarding moment to the element about this same axis1 the strength of the retarding moment being at all times proportional to the velocity with which the element is moving in the course of its vibration4 ,the opposition exercised by the retarding force is not1 however1 a direct one4 ,the motion of the sensitive element arises from the fact that the spinning wheel1 the axle1 and the pendulous weight are tilted about the horizontal axis ,e ,f4 ,the retarding force is applied about the vertical axis1 not because that is the axis on which the sensitive element is oscillating1 but because a force so applied produces a movement about the horizontal axis tending to wipe out the tilt and so eliminate the cause of the oscillation4 ,illustration3 ,,fig4 #bb4 ,gyro-,pendulum with ,axle ,tilted4 ,an alternative method of damping the oscillation is possible4 ,the effectiveness of the pendulous weight may be damped by virtually reducing the weight of the pendulum bob instead of reducing the tilt at which it is acting4 ,in ,fig4 #bb the wheel of a gyro-pendulum system is shown tilted4 ,the weight ,s is trying to turn the wheel on the horizontal axis ,e ,f in the direction ,r1 and as a consequence is causing precession about the vertical axis ,h ,j in the direction ,t4 ,instead of attempting to damp this precession by applying a force about the vertical axis so as to reduce the tilt1 we might virtually reduce the weight ,s by applying an upward force ,w at the end ,b of the axle1 or1 what is the same thing1 a downward force ,v at the other end4 ,such a force1 if it were acting alone1 would tend to turn the wheel in the direction opposed to that of the arrow ,r1 and would consequently cause the wheel to process on the vertical axis in the direction opposed to that of the arrow ,t4 ,acting in conjunction with the weight ,s1 the force ,w would thus damp the natural motion by endeavouring to produce a counter precession of the wheel about the vertical axis4 ,this alternative method of damping the oscillations is adopted in the ,brown gyro-compass4 ,the mechanism of the ,brown method is shown diagrammatically in ,fig4 #bc4 ,the wheel1 as in the early ,anschtz compass1 runs in an enclosing case1 and acts as a blower1 developing a pressure of about #c in4 of water4 ,at each side of the casing a bottle ,l1 ,m is attached to the housing of the axle4 ,a pipe ,n connects the bottoms of the bottles1 while a second pipe ,p joins their upper ends4 ,at its mid point the pipe ,p is interrupted1 and a box ,q is inserted in the gap4 ,this box1 as shown separately1 is open on the underside1 and is provided with a central partition4 ,the air from the casing is delivered through the hollow trunnion ,f1 and passes upwards into the box ,q from the orifice ,r4 ,when the axle is horizontal the air blast entering the box ,q is divided into two equal portions by the central partition1 and exerts an equal pressure inside the bottles4 ,these two bottles are half-filled with oil1 which under the equal air pressures and the horizontal configuration of the system lies at an equal level in each bottle1 and does not therefore unbalance the sensitive element on the axis ,e ,f4 ,if1 however1 the axle tilts the balance is disturbed4 ,thus if the end ,b rises the box tilting with the wheel will assume some such position relatively to the nipple ,r--which is free of the trunnion ,f--as that shown at ,t4 ,the central partition will divide the air jet unequally1 and a greater pressure will therefore be exerted inside the bottle ,l than in the bottle ,m4 ,thus the oil level in ,l falls1 while the oil level in ,m rises4 ,the difference in the weights of the two bodies of oil acts as a turning moment tending to rotate the casing about the horizontal axis ,e ,f in the direction ,u--that is to say1 in the direction opposed to that in which the pendulum weight ,s is trying to turn the casing4 ,since the pendulum weight causes precession about ,h ,j in the direction ,v1 the weight of the unbalanced oil will cause or tend to cause precession about ,h ,j in the reverse direction4 ,illustration3 ,,fig4 #bc4 ,damping ,system of ,brown ,compass4 ,should the end ,b of the axle fall1 the air jet will exercise excess of pressure in the bottle ,m1 the oil level will rise in the bottle ,l1 and as a consequence will tend to make the system precess about ,h ,j in the direction of the arrow ,v4 ,thus in both cases the precession which would be produced by the unbalanced weight of oil if it acted alone is in the reverse direction to that produced by pendulum weight when the axle tilts under the rotation of the earth4 ,as the excess of pressure in either bottle varies with the ratio in which the central partition of the box ,q divides the air blast1 and as that ratio varies with the angle of tilt1 it follows that the opposition of the applied precession increases with the angle of tilt1 and therefore with the rate at which the sensitive element is turning on ,h ,j under the influence of the pendulum weight ,s4 ,the system therefore fulfils the requirements of a satisfactory damping method4 ,it may be noted that were the air blast and the box ,q suppressed and the pipe ,p made continuous1 a tilt of the axle would cause oil to flow from one bottle to the other in the endeavour of the liquid to preserve its surface horizontal4 ,this flow might be constricted by means of a valve in the pipe ,n1 and would then serve to damp the oscillations1 for the arrangement would constitute virtually an oil dashpot4 ,this plan would not prove satisfactory in practice1 because the tilts which are attained are so small that the amount of oil flowing between the bottles would be insignificant4 ,further1 the flow would almost certainly lag behind the movement to be damped4 ,the application of the air blast to the surface of the oil greatly increases the rate of flow of the oil and correctly synchronises it with the tilt4 ,the extent to which the damping effect is allowed to act on the system can be regulated by means of a needle valve in one of the bottles1 which valve is made to control the orifice in the bottom of the bottle through which the pipe ,n enters4 ,the damping system adopted in the latest form of ,anschtz compass cannot very conveniently be described at this point4 ,it will be dealt with later on in ,chapter ,,xvii4 ,for the present it is sufficient to say that the system used is in principle very similar to that found in the ,brown compass4 ,,chapter ,,viii ,,the ,,latitude ,,error ,having explained the necessity for damping the horizontal vibrations of the gyro-compass axle and how the damping force is generated and applied in the ,anschtz1 ,sperry1 and ,brown compasses1 we are now in a position to discuss the various errors to which the gyro-compass is open and how those errors are in practice eliminated1 reduced1 or allowed for4 ,the first error to which we shall turn our attention is that known as the latitude error4 ,this error is a direct consequence of the necessity we are under of damping the horizontal vibrations of the compass axle4 ,it will be recalled that in any latitude other than the equator the compass rests on the north with a slight tilt up or down of the axle1 the tilt increasing progressively as the poles are approached and being upwards in northern latitudes and downwards in southern4 ,the tilt acquired in any latitude is perforce and automatically just sufficient to precess the axle westwardly or eastwardly at the rate required at that latitude to keep the axle on the north once it has found it4 ,let us suppose1 then1 that a compass of the early ,anschtz type is stationed in some degree of northern latitude4 ,the axle in acquiring the appropriate tilt will clearly disturb the equilibrium between the two sections of the air blast4 ,as the axle tilts its north end ,b upwards the north section of the air blast will enlarge and the southern diminish in area4 ,thus the tilting of the axle brings into play an air blast reaction which is applied as a moment tending to turn the sensitive element about the vertical axis ,h ,j in such a way as to make the north end move eastwardly4 ,the actual movement produced will1 in accordance with the fundamental rule1 be a precession of the axle about the horizontal axis1 the end ,b going down4 ,thus the tilt upwards of the axle required for the correct working of the compass is opposed4 ,it is not allowed to acquire the full value which it should have1 but instead stops at the point at which a balance is struck between the upward moving tendency of the axle resulting from the earth's rotation and the downward moving tendency caused by the uneven air blast reaction4 ,the westerly precession of the axle about the vertical axis is now that arising from the reduced angle of tilt1 and is less than the rate required to keep the end ,b pointing to the north4 ,the resulting deviation is known as the latitude error4 ,its magnitude depends upon the latitude and upon such factors as the speed of the wheel1 its efficiency as a blower and other items incidental to the particular design of the compass4 ,in the ,sperry compass the latitude error arises from an exactly similar cause4 ,thus the tilt of the axle is transmitted through the excentric pin to the casing as a moment about the vertical axis ,h ,j1 tending to make the north end of the axle move over to the east1 and therefore causing the axle to precess vertically downwards until the position of balance is reached4 ,in the ,brown compass1 on the other hand1 it is claimed that there is no latitude error4 ,thus when in northern latitudes the north end of the axle tilts upwards under its unsatisfied desire to set itself parallel with the earth's polar axis1 the air blast is divided unequally between the two partitions of the box4 ,excess pressure is exerted inside the damping bottle attached to the northern face of the wheel casing1 and as a result an excess of oil accumulates in the damping bottle on the south side4 ,the weight of the excess oil applies a turning moment to the spinning wheel about the horizontal axis1 and tends to lift the north end of the axle upwards in opposition to the pendulum weight1 which tends to turn it downwards about the same axis4 ,thus in this compass the natural tilt of the axle is not interfered with by the action of the damping arrangement1 for that action is applied about the horizontal axis ,e ,f1 and not1 as in the ,anschtz and ,sperry compasses1 about the vertical axis ,h ,j4 ,the damping action in the ,brown compass is virtually equivalent to a subtraction from the weight of the pendulum bob1 and is not applied1 as in the other compasses1 to the reduction of the tilt of the axle4 ,it tends1 therefore1 to reduce the natural rate of westerly precession required at the given latitude to keep the axle on the north4 ,actually1 however1 the tilt acquired by the axle will take cognisance of the reduction in the weight of the bob caused by the weight of the excess oil4 ,the north end of the axle will rise until the effective moment of the bob--that is1 the moment due to its own actual weight less the moment due to the excess oil in the southern damping bottle--is such as will automatically generate the required rate of westerly precession for the given latitude4 ,in the ,anschtz and ,sperry compasses the latitude error can be allowed for by shifting the position of the lubber line relatively to the ship's longitudinal centre line in accordance with calculated tables of the error4 ,the exact value of the error depends upon the design of the compass1 but in both types it increases with the latitude4 ,analysis shows that--at least in the early ,anschtz compass--the latitude error may be neglected if the change of latitude is less than #aj deg4 ,in the ,sperry compass1 as we shall see presently1 the movement of the lubber line to correct the latitude error is effected by a very pretty piece of mechanism1 which also can control the position of the lubber line to allow for a second error to which the compass is open1 to wit1 the so-called 8north steaming error40 ,in some respects as an alternative to moving the lubber line1 the latitude error may be completely eliminated for a particular latitude--the latitude of most frequent use would naturally be chosen--by attaching a small weight to the north side of the wheel casing4 ,in the early ,anschtz compass such a weight was provided as indicated at ,t in ,fig4 #af4 ,this weight and its position were so chosen that at #ej deg4 north of the equator its moment about the horizontal axis ,e ,f1 when the axle of the compass was horizontal1 was just sufficient to supply to the wheel the amount of westerly precession required to keep the axle pointing to the north4 ,the axle being horizontal in that latitude1 the air blast was equally divided by the pendulum shutter1 and as a result the rate of westerly precession produced by the weight ,t was not affected by the existence of the damping system4 ,at other latitudes1 of course1 the latitude error arose1 and at the equator--where ordinarily there should be no latitude error--and south of the equator1 the error became worse than it would have been without the addition of the weight1 unless the weight and its position were adjusted4 ,with the compass corrected for #ej deg4 ,n4 latitude the error had the following values3 ,latitude ,error #fj ,n4 4#f ,e4 #ej ,n4 ,zero #dj ,n4 4#e ,w4 #bj ,n4 #a4a ,w4 #j #a4f ,w4 #bj ,s4 #b4a ,w4 #dj ,s4 #b4g ,w4 #fj ,s4 #c4h ,w4 ,in the ,sperry compass no attempt is made to eliminate the latitude error at any particular latitude1 the correction applied to the lubber line being solely relied upon to allow for it4 ,in this compass1 therefore1 the latitude error is zero at the equator4 ,at #ej deg4 north or south it amounts to #b deg41 easterly and westerly respectively4 ,a feature found in this type of gyro-compass of a connected nature calls1 however1 for mention at this point4 ,this feature consists of mounting the pivots ,m ,n 7,fig4 #ai71 on which the bail swings1 within excentric housings and graduating the edge of the housings with a scale of latitudes4 ,in this way the pivots can be displaced to one side or the other of the vertical plane containing the axis ,h ,j by an amount proportional to the latitude4 ,when the spinning wheel is at rest1 this displacement makes the angle between the axle of the wheel and the plumb line through the centre of gravity of the bail a little less or a little more than a right angle1 and in northern latitudes makes the north end of the axle dip below the horizontal and in southern latitudes rise above it4 ,the scale is so graduated that the dip of the axle or its rise when the wheel is not running is just equal to the natural rise or dip which the axle would acquire at any given latitude with the wheel running and with the bail pivots in the mean position4 ,as a result1 when the compass is in service and the bail pivots are adjusted for the latitude of the station1 the natural rise or dip of the axle leaves the axle horizontal1 but deflects the bail weight from the vertical by the amount required to generate the correct degree of easterly or westerly precession4 ,thus in the ,sperry compass the axle1 if the latitude bail correction is applied1 is at all latitudes horizontal when resting on the north4 ,several advantages are thus secured1 the chief being that the effect of any change in the speed of the spinning wheel or of a complete failure of the electric supply driving the wheel is greatly reduced or spread over a longer interval4 ,were the axle as well as the bail allowed to acquire the rise or dip proper to the latitude1 the axle during a change of speed of the wheel would tend to deviate and develop an error which might prove misleading4 ,if1 however1 the axle is not allowed to acquire the rise or dip appropriate to the latitude1 the error introduced by a change in the speed of the wheel takes longer to manifest itself1 and is reduced in magnitude4 ,as it is difficult to guarantee that the voltage of the current supplied to the compass will not vary1 this feature of the ,sperry compass is undoubtedly of practical advantage4 ,,chapter ,,ix ,,the ,,north ,,steaming ,,error ,the source of error which we have just discussed affects the gyro-compass whether it is on land or on a ship4 ,we have now to discuss certain errors which are only met with when the compass is mounted on board a moving ship4 ,the first of these errors to which we will refer is sometimes called the 8north steaming error10 although it is equally associated with a due south course4 ,imagine the vessel on which the gyro-compass is fitted to be sailing due east along the equator4 ,if its speed is1 say1 #bj knots it would1 if it could1 circumnavigate the globe in #de days4 ,its velocity round the earth's axis apart from the rotation of the earth is thus #j4jjjjae revolution per minute4 ,as the speed of the earth on its polar axis is #j4jjjg revolution per minute1 the actual rate at which the gyro-axle is being carried round in space is #j4jjjgae revolution per minute4 ,if the vessel is sailing due west its speed opposes that of the earth1 so that the actual rate at which the gyro-axle is being carried round in space is #j4jjjfhe revolution per minute4 ,as compared with the same gyro-compass on land1 the only effect of the ship's speed on these courses is in the one case to increase the magnitude of the directive force and in the other case to reduce it1 the increase and reduction both being quite small--about #b per cent4 actually4 ,sailing due east or west in latitude #fj deg4 north or south1 its speed would cause the vessel to circumnavigate the globe in #bb days4 ,on these courses in either of these latitudes1 therefore1 the directive force would be increased or diminished respectively by about #d per cent4 ,illustration3 ,,fig4 #bd4 ,the ,north ,steaming ,error at #j ,deg4 and #fj ,deg4 ,n4 ,if1 now1 the vessel starts at the equator and sails due north1 its speed is at right angles to the speed with which the rotation of the earth is carrying the gyro-compass round in space4 ,the speed of the ship--#bjbf ft4 per minute--may be represented by ,a ,b 7,fig4 #bd7 and the speed of rotation of the earth--#ib1djj ft4 per minute--by ,a ,c4 ,the actual speed and direction in which the gyro-compass is being carried round in space is ,a ,d1 and the actual axis about which it is being carried round is not the earth's polar axis ,n ,s1 but an axis ,n ,s at right angles to ,a ,d4 ,the gyro-axle will settle1 therefore1 on the line ,n ,s1 and not on the true north and south meridian4 ,the true north will be to the east of the indicated north by the angle ,n ,a ,n1 which for the ship speed in question--#bj knots--will be #a4be deg4 ,if the ship starting from the equator sails due south1 the deviation will be towards the opposite side1 the true north lying #a4be deg4 to the west of that indicated by the compass4 ,if the course is neither due north or south nor due east or west1 the deviation will have some intermediate value between zero and #a4be deg41 the true north lying to the east of the indicated on all courses with a northern component and to the west on all courses with a southern component4 ,if the ship is in latitude #fj deg4 north1 and is steaming north1 its speed will carry it1 as before1 #bjbf ft4 northward per minute1 as at ,e ,f1 but1 as its distance from the earth's polar axis is now only half what it was at the equator1 the earth's rotation is carrying it round at only half the equatorial speed1 namely1 with a velocity ,e ,g of #df1bjj ft4 per minute4 ,the compass is therefore being carried round in space with a velocity represented in magnitude and direction by ,e ,h--that is to say1 it is being rotated not actually about the earth's polar axis ,n ,s1 but about an axis ,n ,s at right angles to the resultant velocity ,e ,h4 ,the axle will therefore align itself parallel with ,n ,s1 and not with ,n ,s4 ,thus in that latitude at the given speed the true north will lie #b4e deg4 eastward of that indicated by the compass4 ,on north-easterly or north-westerly courses at #fj deg4 north latitude the deviation will lie between zero and #b4e deg41 while on all courses with a southerly component the true north will be to the west of the indicated north by some amount between the same two limits4 ,our demonstration is not strictly accurate1 for1 arising from the rotundity of the earth1 the speed ,e ,f of the ship is not really in the plane of the paper1 but should be drawn inclined with the end ,f below the level of ,e4 ,accurate analysis1 however1 shows that the deviation arrived at by neglecting this fact is substantially correct4 ,in the tables issued in connection with the early ,anschtz compass the deviation at #bj knots due north or south in latitude #fj deg4 ,n4 is given as #b4e deg41 while those issued by the ,sperry ,company give it1 by interpolation1 as #b4da deg4 ,it will be seen that this error is a natural one--that is to say1 it is not caused by any peculiarity in the design of the compass in use4 ,the table of deviations on various courses1 at various speeds1 in various latitudes1 compiled for one type of compass would be equally applicable to any other design4 ,in the case of the early type of ,anschtz compass the 8north steaming error0 was corrected solely by means of such tables--that is to say1 to find the exact bearing on which the ship was sailing the compass was read1 and from its reading there was subtracted1 or to it there was added1 arithmetically1 the appropriate figure taken from the tables1 the uncorrected reading being a sufficiently accurate indication of the true course to permit it to be assumed as the true course for the purpose of extracting the correction from the tables4 ,in the ,sperry design the use of similar tables in a similar way is permissible and is provided for4 ,in addition1 however1 provision is made whereby the tables may be entirely dispensed with and the correction applied by moving the lubber line relatively to the ship's fore and aft direction4 ,in the case of the early ,anschtz compass the lubber line could be moved through #d deg4 on each side of the position in which it was parallel with the ship's fore and aft centre line4 ,this movement was1 however1 for the purpose of correcting the latitude error1 and not the north steaming error4 ,in the ,sperry compass provision is made for moving the lubber line so as to correct for both errors4 ,the latitude error depends upon the design of the compass and the latitude4 ,the north steaming error is independent of the design of the compass1 and is determined by the speed of the ship1 the latitude in which it is sailing1 and the course which it is following4 ,latitude thus comes into both errors1 being the sole variable causing one1 and one of the three variables causing the other4 ,it is eliminated simultaneously for both errors1 by operating one dial4 ,a second dial is set to suit the speed of the ship1 while the third factor in the north steaming error1 the bearing of the ship's course1 is automatically brought into the movement of the lubber line by means of an inclined ring carried round with the compass card4 ,a diagram of the ,sperry correction mechanism is given in ,fig4 #be4 ,the lubber ring on which the lubber line is engraved is shown at ,a4 ,the latitude corrector dial ,b is mounted to rotate about its centre and to read against a fixed scale4 ,it is slotted radially to engage a pin ,c on a rod ,d ,e1 one end of which can slide in a fixed guide ,f1 while the other is pivoted to a link ,e ,g pinned at ,h to the lubber ring4 ,if the end ,g of this link be imagined pivoted to some fixed point it is clear that rotation of the dial ,b will communicate a corresponding rotation to the lubber ring4 ,the proportions of the parts and the positions of the pivots and pins are such that by setting the dial to the latitude in which the vessel is sailing the lubber ring is rotated through the angle by which the local latitude error causes the gyro-axle to deviate in its resting position from the true north and south direction4#c ,illustration3 ,,fig4 #be4 ,sperry ,correction ,mechanism for ,latitude and ,north ,steaming ,errors4 ,if the end ,g of the link ,e ,g were pivoted1 as we have supposed it to be1 to some fixed point1 then the latitude ,error could be applied by itself to the lubber ring4 ,it is not1 however1 so fixed1 but is pivoted to a link1 the end ,j of which carries a pin working in a curved slot extending from the centre of the speed corrector dial ,k to its edge4 ,the radius of this curved slot is struck from the point ,g as centre4 ,it follows1 therefore1 that with the speed corrector dial in the position shown the end ,j of the link ,g ,j may be fixed at any point in the curved slot without affecting the position of the point ,g4 ,if1 however1 the speed corrector dial be rotated about its centre1 the movement of the point ,g will vary in magnitude with the position at which the pin ,j is fixed in the curved slot4 ,this position is set against the scale of speed in knots on the edge of the slot1 but the setting is without effect until the speed corrector dial is rotated from the position shown in the diagram4 ,when it is so rotated a corresponding movement is communicated to the lubber ring through the link ,g ,e1 the end ,e of which is now to be regarded as the fixed fulcrum4 ,the two dials ,b ,k are connected by means of a link ,l ,m of length equal to the distance between the dial centres4 ,the end ,l of this link is pivoted to a fixed point on the dial ,b1 but the end ,m is provided with a pin which works within a slot in the dial ,k4 ,thus the setting of the dial ,b to any latitude will not produce rotation of the dial ,k1 but will merely cause the pin ,m to slide in its slot4 ,if1 however1 the link ,m ,n is moved upwards1 the angle through which such movement will rotate the dial ,k will depend upon the exact position of the pin ,m in its slot--that is to say1 upon the setting of the latitude dial4 ,thus a movement of the link ,m ,n will rotate the lubber ring by an amount proportional to the movement of ,m ,n1 to the setting of the pin ,j in the curved slot of the speed dial1 and to the setting of the latitude dial4 ,the link ,m ,n is pinned to a lever ,p ,q1 the end ,p of which is pivoted to a fixed point4 ,the end ,q carries a roller1 which engages within the flanges of a ring ,r permanently fixed in an inclined position on the phantom ring of the compass4 ,the higher end of this ring is situated directly above the north-seeking end of the gyro-axle1 and the lower end directly above the south-seeking end1 its east and west diameter being aligned horizontally4 ,in the position shown in the diagram the ship is supposed to be sailing due west4 ,under this condition1 as we know1 no north steaming error is involved1 the latitude error only applying4 ,the dial ,b can therefore be set to the latitude in which the vessel is sailing so as to apply the latitude correction by itself to the lubber ring4 ,at the same time the pin ,j can be set to read the ship's speed--#ag knots in the case shown4 ,the movement of the link ,j ,g involved in this setting will1 as we have seen1 not alter the setting of the lubber ring1 nor will the setting of the dial ,b rotate the dial ,k from the position shown4 ,the two settings are1 however1 ready to modify the adjustment of the lubber ring as soon as any movement of the link ,m ,n occurs4 ,such a movement arises if the vessel alters course towards the north or south or intermediately1 for when the course is altered the links1 dials1 and lubber ring move with the ship1 while the compass card and the course corrector ring ,r remain stationary4 ,the roller at ,q therefore rises or falls towards the north or south end of the ring1 and produces a rotation of the dial ,k4 ,the amount of the modification thus produced in the setting of the lubber ring relatively to the ship's centre line will depend upon 7#a7 the setting of the pin ,m--that is to say1 the setting of the latitude dial2 7#b7 the setting of the pin ,j on the speed dial2 and 7#c7 the extent of the turn towards the north or south4 ,in this way the correction for the north steaming error is automatically added to or subtracted from the correction for the latitude error to the correct amount on all occasions when the ship's course is altered so as to have a northerly or southerly component1 the adjustment for the error being automatically wiped out in all latitudes when the ship's course is again brought east and west4 ,the mechanism1 it will be seen1 merely by the setting of two dials1 takes cognisance of three independent variables1 of which one enters into the required adjustment in a twofold way4#d ,the total correction is indicated on the corrector scale ,s attached to the binnacle independently of the lubber ring4 ,it will be seen1 then1 that neither the latitude error nor the north steaming error is in the strict sense of the word eliminated1 but is only allowed for1 automatically or arithmetically1 when the compass is read--that is to say1 no attempt is made to force the gyro-axle to rest in the true north and south position in all latitudes at all speeds on all courses4 ,the crucial angle by which the ship is navigated is the angle between its longitudinal centre line and the true meridian4 ,a constant straight course will be held if this angle is maintained constant4 ,if the indicated meridian on the compass is open to error1 the constancy of the navigating angle may be preserved by introducing a similar error into the degree of coincidence between the lubber line and the longitudinal centre line of the ship4 ,the readings of the master compass may therefore be used for the purpose of maintaining the ship on a given course or for setting it on a new one4 ,but for taking the bearing of some passing object1 the fact that it may not be indicating true north may render the observation somewhat more troublesome than it need be4 ,to overcome the difficulty here involved the repeaters served by the master compass are arranged to indicate true north under all conditions1 provided the latitude and speed dials on the master compass are correctly set4 ,the repeaters are operated electrically1 the first element in the transmission system between them and the master compass being a pinion meshing with the gear wheel attached to the phantom ring beneath the master compass card4 ,this pinion is mounted on the lubber ring1 and therefore rotates with the ship round the card when the course is altered4 ,but1 as it is mounted on the lubber ring1 it also partakes of the movement of this ring relatively to the ship produced by the correction mechanism4 ,thus the net indication transmitted to the repeaters when the ship changes course or latitude or speed is compounded of the relative motion of the master compass card and of the lubber ring4 ,the reading of the repeaters thus shows the true north4 ,a pelorus or dummy compass may be combined with the repeaters to facilitate the taking of bearings4 ,in war vessels the actual navigation of the ship is preferably conducted from a repeater1 the master compass for safety in action being disposed below the armoured deck4 ,the ,sperry compass is peculiarly complete in the provision made for taking account of the latitude and north steaming errors4 ,the ,brown compass1 as we have seen1 is free from latitude error1 so that the north steaming error alone has to be dealt with4 ,this error may be allowed for arithmetically in the usual way by means of tables of corrections to be applied to the readings of the master compass4 ,as regards the repeaters1 the error is eliminated by a method involving the mounting of the repeater cards excentrically1 the excentricity being varied to suit the latitude1 speed1 and course4 ,,chapter ,x ,,the ,,ballistic ,,deflection ,having considered the effect of the ship's speed upon the readings of the gyro-compass1 we have next to discuss the effect of the conditions which arise when the speed is changed4 ,it is clear1 of course1 that if the ship's speed is changed1 say from #bj to #aj knots1 the north steaming error appropriate to the course and latitude will fall to half its value when the new speed has been attained4 ,two questions1 however1 may be asked4 ,what happens in the period during which the speed is changing1 and what time elapses between the attainment of the new speed and the attainment by the compass of the new north steaming error proper to the new speed8 ,if an ordinary simple pendulum were hung from the roof of a railway carriage it would be found that so long as the train was travelling smoothly at uniform speed the pendulum would hang vertically without oscillating just as it would do if the train were at rest4 ,if the train increased its speed the inertia of the pendulum--that is to say1 its tendency to go on moving with the old velocity--would cause the pendulum to assume an inclined position behind the vertical1 the deflection being proportional to the rate at which the train was gathering speed4 ,at the instant at which the train ceased to gather speed1 and again assumed a uniform velocity1 the tendency for the pendulum to remain inclined to the vertical would vanish1 but the deflection it possessed at that instant would result in its acquiring an oscillation which would persist until friction1 etc41 damped it out4 ,if when the pendulum were again steady and hanging vertically1 the train began to slow down1 a similar series of events would occur1 only on that occasion the inertia of the bob would cause the pendulum while the train was losing speed to assume a forwardly inclined position1 the angle of which would be a measure of the rate at which the speed was being reduced4 ,on the attainment of the new uniform speed the pendulum would oscillate for a time1 as before4 ,the gyro-compass is in part a pendulous body1 and on a ship changing speed or acquiring speed from rest or coming to a stop it is open to the action of inertia forces just as is our railway carriage pendulum4 ,as a result the compass during the change of speed may exhibit an error--the ballistic error1 as it is called--on top of the existing north steaming error4 ,since the speed of a ship1 at least of a merchant ship1 cannot be changed very quickly1 and since1 further1 the speed of a ship1 at least on long voyages1 is not liable to be changed very often1 it might be thought that this transient ballistic error could be neglected4 ,it probably could be safely neglected if its effect were strictly confined to the actual period of the change in the speed1 but it is not so confined4 ,we have to bear in mind the subsequent vibration of the railway carriage pendulum4 ,a similar oscillation is set up in the gyro-compass after the new speed is attained4 ,since the period of oscillation of the compass is about #he minutes1 and as two or three complete swings have to be made before the damping arrangements can suppress the vibration1 it follows that the compass will not settle in its resting position again until two or three hours after the new speed has been attained1 although the actual change of speed may have been effected inside five minutes or less4 ,illustration3 ,,fig4 #bf4 ,ballistic ,force on ,compass when ,ship's ,speed ,changes4 ,if the ship is sailing on an east-west course1 as represented in ,fig4 #bf1 any change of speed will tend to cause the pendulum weight ,s to move in the direction ,r or ,t1 according as the change of speed is a decrease or an increase4 ,such a tendency will merely throw stresses on to the journals at ,e ,f and ,h ,j1 or1 if the equivalent of the square outer frame is carried on gimbals1 as it actually is in practice1 then the tendency to rotate will be translated into an actual movement of the sensitive element on the axis of the outer mounting coincident or parallel with the axle ,b ,c of the wheel4 ,such a movement will have no gyroscopic effect on the compass1 for the axle is not subjected by it to any non-parallel displacement4 ,no ballistic deflection will therefore occur on that course4 ,illustration3 ,,fig4 #bg4 ,ballistic ,force on ,compass when ,ship's ,speed ,changes4 ,if the ship changes speed while on a west to east course1 the change is similarly without effect on the compass4 ,strictly speaking1 we ought to say that the change of speed will have no effect on the compass if the ship's course is at right angles to the direction in which the axle of the wheel is pointing4 ,only at the equator would this course be due east or west4 ,in other latitudes the latitude error has to be reckoned with1 so that the particular course on which changes of speed are without effect on the compass must be slightly south or north of due east or west4 ,if1 now1 the ship is steaming due north1 as indicated in ,fig4 #bg1 a reduction of its speed will tend to make the pendulum weight swing forward about the east and west axis ,e ,f in the direction ,p4 ,this tendency is clearly equivalent to the application of an upward force to the end ,b of the axle1 and hence1 as we know1 will cause the wheel to precess1 the end ,b of the axle going eastwards4 ,as this precession continues it is opposed by the ever-increasing restoring moment of the directive force1 so that in the end the axle assumes a position in which the directive force just balances the ballistic deflective force4 ,this ballistic deflection remains constant all the time the ship's speed is falling4 ,when the speed reaches the new steady value the axle slowly oscillates back to the true resting position4 ,it is clear that a reduction of speed on a northerly course and an increase of speed on a southerly produce ballistic deflections in a like direction1 the north end of the axle moving towards the east4 ,an increase of speed on a northerly course or a decrease of speed on a southerly produces a westerly ballistic deflection4 ,on intermediate courses deflections of a like kind1 but of intermediate value1 are caused1 for the north and south component of the change of speed is alone effective in tilting the wheel about the east and west axis ,e ,f4 ,imagine a ship steaming due north at #bj knots and changing its speed to #aj knots during a period of five minutes4 ,as it is steaming north there will be a north steaming error1 the axle pointing westwards of true north by an amount dependent upon the speed of the ship and the latitude in which it is sailing4 ,let ,o ,n 7,fig4 #bh7 be the direction of true north1 and let ,o ,a be the direction in which the axle of the gyro-compass aligns itself when the speed is #bj knots1 the angle ,n ,o ,a being the combined latitude and north steaming errors4 ,let ,o ,b be the resting position for the axle when the ship's speed is #aj knots1 the angle ,n ,o ,b being the latitude error--which has not altered--plus the north steaming error--which is now less because of the reduced speed4 ,as the speed is being reduced1 the ballistic action of the pendulum weight causes the axle temporarily to turn eastwards from the true resting position and to align itself in some direction ,o ,c4 ,when the ship has settled down to #aj knots1 the axle leaving the position ,o ,c vibrates about ,o ,b with amplitudes which are being continually reduced by the action of the damping system4 ,illustration3 ,,fig4 #bh4 ,ballistic ,deflection4 ,the point of importance to notice is that the temporary ballistic deflection ,o ,c is eastward of ,o ,a1 just as is ,o ,b the new resting position for the axle4 ,this result is a general one4 ,had the speed been increased instead of decreased1 the ballistic deflection would have been westward of ,o ,a1 just as would be the new resting position for the axle at the increased speed4 ,similarly1 on due southerly courses or on quadrantal courses the ballistic deflection produced by any change of speed is always in the same direction as that in which the axle moves in passing from the old to the new north steaming error4 ,this being so1 it is conceivable that at least1 under certain conditions1 the ballistic deflection position ,o ,c may coincide with the new position ,o ,b for the axle under the new north steaming error4 ,should such a result be obtained1 the ballistic action will1 on a change of speed occurring1 swing the axle straight away into the new resting position4 ,thus ,o ,c and ,o ,b being coincident1 there will be no tendency for the axle to oscillate about ,o ,b when the speed assumes the new steady value1 so that the axle will swing into the new resting position in a 8dead-beat0 manner4 ,the effect of the ballistic action will thus be confined to the period during which the speed is being changed--five minutes in our example--and will not influence the readings of the compass for a period of anything up to two or three hours after the new speed is reached4 ,the ballistic deflection of a pendulum hung from the roof of a railway carriage is dependent upon the length of the pendulum1 and therefore upon its period of natural vibration4 ,so1 too1 the ballistic deflection of the gyro-compass is dependent upon the period of its vibration about the north and south direction4 ,once the characteristics of the compass are determined1 therefore1 the angle ,a ,o ,c 7,fig4 #bh7 of the ballistic deflection is settled by 7#a7 the direction of the ship's course1 and 7#b7 by the rate at which the ship's speed towards the north or south is being changed4 ,it is not affected by the latitude in which the ship is sailing1 the deflection produced by a change of #aj knots in five minutes being the same at the equator as in #fj deg4 or any other latitude4 ,coming to the difference between the old and the new north steaming errors1 it will be seen from our previous explanation of the errors themselves that the magnitude of the angle ,a ,o ,b is independent of the design of the compass in use4 ,it will vary with 7#a7 the direction of the ship's course1 and 7#b7 the extent by which the ship's speed is changed4 ,in addition1 it will also vary with 7#c7 the latitude in which the ship is sailing4 ,from these considerations it is argued that by suitably selecting the period of vibration the ballistic deflection can be made for one particular latitude just equal to the difference between the north steaming errors for any initial and any new speed on any course4 ,in practice the latitude selected is #dj deg4 north or south4 ,it is found by calculation that the period of vibration which must be given to the compass to secure this dead-beat ballistic deflection is such that in this latitude the compass should oscillate in the same period as would a simple pendulum1 the length of which was equal to the radius of the earth4 ,it is for this reason that all modern gyro-compasses have a period of vibration of approximately #he minutes4#e ,in latitudes other than #dj deg4 north or south the ballistic deflection will not be dead-beat1 but by taking this latitude as the mean the subsequent oscillation after a change of speed is--in mercantile vessels at least--sufficiently small to introduce no error of great importance1 except for purposes of very accurate observation1 in which case the observation1 if possible1 would not be made until the vessel had been running for two or three hours after the last considerable change of speed4 ,it is to be noticed that the ballistic error may arise when the ship is altering course without altering its speed4 ,thus1 if it is sailing north and makes a sharp turn eastwards or westwards1 the speed northwards during the turn falls progressively from a maximum at the beginning to zero at the end of the turn1 even although the actual linear speed of the ship remains constant throughout4 ,turning in either direction is thus equivalent to a deceleration of the ship's northerly speed1 so that an easterly ballistic deflection may be expected in both cases4 ,on the other hand1 a turn towards the north from a due east or west course is equivalent to an acceleration of the ship's northerly speed1 and1 as a result1 a westerly ballistic deflection is produced4 ,in latitude #dj deg4 north or south both these ballistic deflections would be dead-beat in a modern gyro-compass1 being just sufficient in the one case to eliminate and in the other to apply the appropriate north steaming error4 ,in a very recent improvement1 it is understood1 it has been found possible to provide means whereby the ballistic deflection is made dead-beat in all latitudes4 ,,chapter ,,xi ,,the ,,quadrantal ,,error ,let a stone be tied to the end of a string and be flung forward while the free end of the string is held in the hand4 ,let the hand holding the free end be moved forward at first with the same speed as the stone in its flight1 and then let the hand be drawn back4 ,we know that as soon as the string becomes taut--if it were not already fully extended--the hand would feel a pull exerted on it in the direction in which the stone was initially projected4 ,this pull arises from the fact that the stone has been given momentum in the direction of its flight1 and while this momentum is being taken out of it by the backward pull of the hand and new momentum in the backward direction is being communicated to it the stone reacts forcibly on the string and therefore on the hand4 ,in the gyro-compass the pendulum weight may be likened to the stone1 the stirrup carrying the weight to the string1 and the spinning wheel to the hand4 ,while the ship is steaming at a uniform speed1 the system is in the condition existing just after the stone has been projected from the hand1 and while the hand is following it with equal speed4 ,a change in the ship's speed--a reduction of its speed to be quite correct in our analogy--is comparable with the drawing back of the hand4 ,the pull of the stone exerted on the hand at this instant is represented by the tendency of the weight to continue moving at the former speed1 and the resultant 8kick0 which by its attempt to do so it applies to the spinning wheel4 ,this kick is communicated to the spinning wheel through stiff members and rigid connections1 and not through a flexible string4 ,it really acts at the centre of gravity of the pendulum weight1 and is therefore felt by the spinning wheel1 not as a straight pull1 but as a force tending to turn the wheel about the horizontal axis ,e ,f4 ,the kick of the weight when the ship changes speed on any course having a north or south component produces1 as we have seen1 a transient ballistic error which may influence the correctness of the compass readings for some hours after the change of speed has been completed4 ,we have now to show that similar kicks occur under other conditions1 and may similarly affect the accuracy of the readings4 ,the subject which we are about to discuss is the so-called quadrantal error which1 unless steps are taken to eliminate it1 appears in the compass readings when the ship is sailing in a rough sea on any course other than due north1 south1 east1 or west4 ,it is caused by the rolling and pitching of the ship4 ,the efforts made to eliminate it have influenced the evolution of the gyro-compass to a greater extent than probably those directed towards overcoming or allowing for all the other errors combined to which the device is open4 ,illustration3 ,,fig4 #bi4 ,effect of ,rolling on due ,north ,course4 ,the gyro-compass on board ship is usually1 although in no way necessarily1 mounted above the vessel's metacentres--transverse and longitudinal--about which rolling and pitching take place4 ,in ,fig4 #bi we show a section1 looking aft1 of a vessel steaming due north with a gyro-compass mounted on its deck4 ,it is clear that1 so far as the weight ,s is concerned1 it may be regarded when the ship rolls as the bob of an inverted pendulum vibrating in an east and west plane1 through an angle equal to the angle of the ship's roll4 ,from what we have already said1 it will be seen that at the end of a roll to port the hand holding the string attached to the flying stone1 hitherto following it1 begins its withdrawal to the east4 ,the weight checked in its movement to the west communicates a kick to the sensitive element4 ,this kick1 a westward force1 is applied at the centre of gravity of the weight1 and clearly does nothing more than throw a stress on to the journals at ,e ,f and ,h ,j1 and through the square frame on to the deck4 ,it does not therefore affect the direction in which the gyro-axle is pointing4 ,if1 as is actually the case in practice1 the equivalent of the square frame is not fixed directly to the deck1 but is mounted inside the binnacle on athwartship and longitudinal gimbals1 the square frame will swing on the longitudinal gimbal axis4 ,the weight ,s will1 however1 still act during the ship's roll as an inverted pendulum1 as at ,z ,z1 in which the bob remains more or less parallel with its original direction4 ,under these conditions the kick of the weight at the end of a roll to port will rotate the frame in the direction of the curved arrow about the longitudinal gimbal axis4 ,as this axis is coincident or parallel with the gyro-axle1 the gyro-axle is not moved otherwise than parallel with itself either by the actual roll of the ship or by the kick of the weight at either out position4 ,thus a ship steaming north may roll with impunity without affecting the accuracy of the direction in which the gyro-axle is pointing4 ,illustration3 ,,fig4 #cj4 ,effect of ,rolling on due ,west ,course4 ,it is clear that rolling on a due south course or pitching on a due east or west course is similarly without effect on the gyro-compass4 ,now consider the case of the ship rolling when on a due west course 7,fig4 #cj74 ,it is clear that here again the weight ,s may by itself be considered as the bob of an inverted pendulum vibrating through the angle of the ship's roll1 this time in a north and south plane4 ,the weight ,s has1 in addition1 its own pendulum action on the horizontal axis ,e ,f1 and tends to keep the gyro-axle horizontal throughout the roll4 ,at the end of a roll to port the kick of the weight ,s1 a southerly force acting at the centre of gravity of the weight1 tends to turn the spinning wheel in the direction ,r about the horizontal east and west axis ,e ,f4 ,this kick is clearly equivalent to the application of an upward force at the end ,c of the axle or a downward force at the end ,b1 and therefore1 as we know1 will cause the axle not to turn about ,e ,f1 but to precess the north end ,b towards the west about the axis ,h ,j4 ,the roll to starboard now takes place1 and at its end the kick of the weight occurs in the northerly direction4 ,this kick will tend to turn the spinning wheel about the east and west axis ,e ,f in the direction ,t1 and is clearly equivalent to the application of an upward force at the end ,b of the axle4 ,such a force1 as we know1 will cause the end ,b of the axle to precess towards the east4 ,it will thus be seen that the western deflection of the axle produced by the precession which occurs at the end of the roll to port is counterbalanced and automatically eliminated by the eastern deflection produced by the precession occurring at the end of the roll to starboard4 ,the only effect on the compass caused by the rolling of the ship is therefore a vibration of the axle about the north and south direction4 ,we should really say that the only effect is the application to the sensitive element of a vibratory influence in tune with the rolling of the ship4 ,as the ship's period is very small compared with that of the compass about the axis ,h ,j--about #e to #ab seconds as compared with about #he minutes--this vibratory influence practically fails to disturb the steadiness with which the axle points to the north4 ,it is clear that pitching of the ship when on a due west course has the same effect--or lack of effect--on the compass as rolling on a due north course1 and that pitching on a due north course has the same effect as rolling on a due west course4 ,it is further clear that due south and due east courses are similar in this respect to due north and due west courses4 ,thus when the ship is on a cardinal course neither pitching nor rolling disturbs the axle of the spinning wheel from its north resting position4 ,matters are quite otherwise1 however1 when the course is an inter-cardinal1 or quadrantal1 one4 ,the discovery of this unlikely fact was in large part due to the investigations of the ,compass ,department of the ,british ,admiralty4 ,it does not appear to have been known1 or at least its full significance does not seem to have been appreciated1 in the earlier days of gyro-compass construction4 ,at least one early design of compass--the ,anschtz of #aiaj--;did not include means of eliminating or allowing for the 8quadrantal error10 and soon became obsolete as a result primarily of the omission4 ,that the later design of ,anschtz compass successfully overcomes the difficulties introduced by this error is shown by the admitted excellence of the navigation of the ,german submarines1 which vessels were universally fitted with compasses of this design1 and which1 like all their class1 suffer much from rolling and pitching4 ,illustration3 ,,fig4 #ca4 ,external ,gimbal ,mounting4 ,so far we have been able to use a very simple model to demonstrate the properties and errors of the gyro-compass4 ,we have now reached a subject which1 if it is to be explained correctly1 requires us to adopt a model of a more elaborate nature1 one in which the outer square frame of our simple model is itself mounted on a pair of gimbal axes4 ,we have already referred to this type of mounting in our second chapter and above in connection with ,fig4 #bi4 ,in ,fig4 #ca we show it with the pendulous weight added to the sensitive element4 ,we may take it that the axis ,t ,u1 about which the entire compass system may swing1 is parallel with the longitudinal centre line of the ship1 and that the frame ,y is fixed to the deck4 ,illustration3 ,,fig4 #cb4 ,effect of ,rolling on a ,due ,north ,course 7,simple ,mounting74 ,illustration3 ,,fig4 #cc4 ,effect of ,rolling on a ,due ,north ,course 7,external ,gimbal ,mounting74 ,in order to make quite clear the difference between the two forms of mounting1 we give in ,figs4 #cb and #cc two corresponding sets of views showing the compass system on a ship while steaming due north and rolling4 ,with the simple mounting 7,fig4 #cb7 the weight ,s constantly remains radially below the centre of the spinning wheel1 and the kicks which it applies at the out port and out starboard positions merely stress the compass mountings4 ,with the more elaborate mounting 7,fig4 #cc7 the weight tends to remain constantly in the vertical below the centre of the spinning wheel1 but the kicks occurring at the out positions1 if the rolling is continued for any length of time1 cause it to swing beyond the vertical1 so that as the ship rolls the whole compass system acquires an oscillation on the axis ,t ,u 7,fig4 #ca7 in tune with the rolls4 ,it is to be noted that the period of vibration of the compass system about the axis ,t ,u is very much less than that of the sensitive element about the axis ,h ,j4 ,the latter1 as we have seen1 is about #he minutes1 and is determined1 in part at least1 by the high speed of the spinning wheel4 ,about ,t ,u the vibration1 however1 does not call any gyroscopic force into play1 for it takes place without causing the axle of the spinning wheel to alter its direction4 ,the period of this vibration is therefore not affected by the speed of the spinning wheel4 ,it may be quite small--from one to two seconds--and therefore comparable with the ship's rolling period4 ,if1 then1 no steps are taken to prevent it1 the compass system as a whole will acquire a swing on the axis ,t ,u 7,fig4 #ca7 when the ship is steaming north--or south--and rolling4 ,its period of vibration on this axis is not strictly that which it would have were it set oscillating about the axis ,t ,u with this axis mounted on a fixed frame1 for the vibration is not a free one1 but is a forced vibration under the impulses communicated by the rolling of the ship4 ,whatever may be the exact free period of swing of the compass system on the axis ,t ,u1 so long as it is somewhere near that of the ship's roll or a small sub-multiple of the rolling period1 the tendency is for the compass swings to settle down in such a way that1 as represented in ,fig4 #cc1 the pendulum weight ,s reaches its extreme out positions simultaneously with the ship's arrival at its extreme port and starboard heels4 ,in the course of one complete roll of the ship1 however1 the compass system may make more than one complete vibration about ,t ,u4 ,it is clear that whether the bob ,s remains radially below the centre of the wheel1 as in ,fig4 #cb1 or remains vertically below it1 or swings past the vertical1 as in ,fig4 #cc1 the effect of rolling when the ship is on a due north course is as established in connection with ,fig4 #bi4 ,either a tendency to rotate the wheel about an axis coincident with its axle or an actual rotation of the wheel about this axis results from the kicks of the weight ,s at the out positions4 ,whichever it is1 there is no gyroscopic effect called into play1 and the axle is not subjected to any influence causing it to move away from the due north direction4 ,we have now to discuss what happens if the vessel rolls or pitches when sailing on an inter-cardinal or quadrantal course4 ,in ,fig4 #cd we show in plan a vessel steaming on a north-west course4 ,when the ship rolls the compass oscillates in the path ,a ,d1 a path1 that is to say1 at right angles to the ship's centre line and curved upwardly by reason of the fact that the compass is mounted above the rolling centre of the ship4 ,the axle of the compass during this movement tends to point steadily towards the north under the action of the directive force4 ,the wheel therefore oscillates from ,a to ,d with its axle askew relatively to the path of oscillation1 and not at right angles to it1 as in ,fig4 #bi1 or parallel with it1 as in ,fig4 #cj4 ,the oscillation ,a ,d can1 however1 be resolved into two components1 namely1 a north and south oscillation ,d ,k1 in which the axle is parallel with the path1 and an east and west oscillation ,k ,a1 in which it is at right angles to the path1 both paths being curved upwardly4 ,illustration3 ,,fig4 #cd4 ,ship ,rolling on ,n4,w4 ,course4 ,the oscillation ,d ,k1 taken by itself1 is clearly exactly equivalent to that shown in ,fig4 #cj4 ,the southward kick of the weight at the 8out position0 ,d 7,fig4 #cd7 just neutralises the disturbing effect of the northward kick at the 8out position0 ,k4 ,similarly1 the oscillation ,k ,a1 taken by itself1 is exactly equivalent to that shown in ,fig4 #bi or its modification1 ,fig4 #cc4 ,the eastward kick of the weight at ,a and the westward kick at ,k cannot do more than turn the mounting round the axle of the spinning wheel4 ,they do not tend to change the direction of the axle4 ,hence1 taken separately1 each component oscillation ,d ,k and ,k ,a is without disturbing effect upon the direction in which the axle is pointing4 ,taken together1 however1 their united effect is not the sum of their separate effects1 as we might hastily assume4 ,we must not suppose that the north and south kicks proper to the north and south component of the oscillation are without effect upon the east and west component or vice versa4 ,at the out position ,d of the component oscillation ,k ,d the weight is not only kicked southwards1 but also receives a westward kick from the other component oscillation ,a ,k4 ,at ,k it receives both a northward kick and an eastward kick4 ,similarly1 at the out position ,k of the component oscillation ,a ,k the weight is kicked not only to the west but also to the south1 by virtue of the oscillation from ,k to ,d4 ,at the other out position ,a of this component the weight is kicked eastwardly1 and also to the north4 ,if we take cognisance of these additional kicks1 we may treat the component oscillations separately and add the separate results to get the true effect of the actual oscillation ,a ,d4 ,returning1 then1 to ,fig4 #cj1 let us suppose that in the oscillation there shown1 the equivalent of the component ,k ,d1 we apply a westward kick to the weight ,s when the ship reaches its port out position--that is to say1 a kick out of the paper--and that at the starboard out position it receives an eastward kick into the plane of the paper4 ,it is clear that these additional kicks do nothing more than tend to rotate the wheel about its axle or an axis coincident or parallel with its axle4 ,they cannot therefore call any gyroscopic effect into action1 and as a result do not disturb the direction in which the axle is pointing4 ,the component oscillation ,d ,k1 even with the extra kicks added1 is thus harmless4 ,taking ,fig4 #cc rather than ,fig4 #bi as the equivalent of the component oscillation ,a ,k1 let us suppose that at the port out position the weight is subjected to an additional kick to the south--that is1 into the plane of the paper--and that at the starboard out position it receives an extra kick to the north1 or out of the plane of the paper4 ,the southward kick on the weight at the port out position will tend to cause the end ,b of the axle to turn down towards ,j1 but1 in accordance with the fundamental gyroscopic rule1 the actual motion of the sensitive element will not be a rotation about the axis ,e ,f1 but a precession about the axis ,h ,j1 the end ,b of the axle moving in the direction ,k4 ,similarly1 the northward kick at the starboard out position will tend to cause the end ,b of the axle to rise towards ,h1 and will therefore produce precession in the direction ,l4 ,now these two precessional movements1 ,k and ,l1 can be resolved into horizontal and vertical components ,n ,m and ,q ,p respectively1 as shown separately1 and of these components it is clear that the movement ,n cancels the movement ,q4 ,on the other hand1 the two components ,m and ,p are in the same direction1 and therefore do not cancel each other4 ,their effect is cumulative1 and with each succeeding roll the end ,b of the axle tends to rise higher and higher in the true vertical plane ,b ,r4 ,it may be explained1 perhaps1 that this motion is possible1 in spite of the fact that there is no actual axis in the mounting of the compass at right angles to the true vertical plane1 except at the instant when the compass is passing through the even keel position in the course of its swing from side to side4 ,as the rolling of the ship starting from zero increases up to a more or less steady value1 the end ,b of the axle rises slowly at first and then at a constant rate1 as exaggerated at ,t4 ,such of this upward movement as occurs while the compass is passing through the even keel position can be effected by means of a rotary motion purely about the axis ,e ,f4 ,elsewhere in the swing of the compass from side to side it is accommodated by a partial rotation on ,e ,f and a partial rotation about ,h ,j4 ,these two axes between them permit the end ,b of the axle to rise in a vertical plane at any point of the compass swing just as effectively as would an axis which was at right angles to the vertical plane ,b ,r4 ,to appreciate the significance of this vertical rise of the north end ,b of the axle1 let us consider the condition of the compass as it is passing through the even keel position4 ,the elevation of the end ,b out of the horizontal plane causes the weight ,s to swing forward towards the north1 and therefore to apply a turning moment to the spinning wheel about the axis ,e ,f4 ,this turning moment will tend to bring the end ,b down again to the horizontal1 but1 as we know1 the actual motion produced will be a precession about the axis ,h ,j1 the end ,b moving westwards4 ,looking at ,fig4 #cd1 it will thus be seen that the net effect of the rolling of the ship is to deviate the axle of the compass in the direction shown at ,v--that is to say1 in the direction required to set the plane of the spinning wheel parallel with the plane of the rolling by the shortest possible course--in our example by a rotation westward through #de deg4 ,with the axle so deflected the earth's rotation1 of course1 calls a directive force into play1 tending to restore the axle to the north and south line1 but if the ship is rolling violently the deviating force will be much stronger than the directive force until a very considerable angle of deflection is reached4 ,when the balance is struck1 the axle settles down with a steady deviation towards the west1 which will remain constant as long as the rolling is maintained4 ,in practice the violence of the rolling varies almost from roll to roll1 for it represents a conflict between the natural period of rolling of the ship and the period of the waves4 ,as a consequence1 the deviation of the compass varies somewhat during the rolling1 but it is always westwardly if the course is in the north-west quadrant4 ,a little consideration will show that if the ship is steaming towards the north-east the additional kick on the weight ,s 7,fig4 #cc7 at the port out position will be towards the north and at the starboard out position towards the south4 ,the end ,b of the axle is therefore precessed vertically downwards instead of upwards1 and as a result the deviation of the axle is eastwardly4 ,in general1 if the ship is steaming on any course in the north-west or south-east quadrant1 the deviation caused by her rolling will be towards the west4 ,it will be towards the east if the course is in the north-east or south-west quadrant4 ,the effect of pitching on a quadrantal course is to cause a deviation of the axle in the direction opposed to that of the deviation caused by rolling1 so that if the vessel be both rolling and pitching1 the deviation is somewhat less than it would be if rolling only had to be considered4 ,it may perhaps be thought that the quadrantal error is an effect produced by the double gimbal mounting of the compass1 and that had we adhered to our simple model1 as shown in ,fig4 #cb1 it would not arise4 ,this is not so4 ,if ,fig4 #cb be taken as representing1 after the manner of ,fig4 #cc1 the north view of the compass on a ship steaming due north-west1 it will be seen that the southern kick at the port out position causes the axle to precess in the direction ,k and the northern kick at the starboard out position in the direction ,l4 ,resolving these movements as before1 we see that1 while the horizontal components ,n ,q again cancel1 the vertical components ,m ,p are1 as in ,fig4 #cc1 cumulative in effect1 but this time they result in the end ,b of the axle moving downwards1 and therefore finally cause it to precess eastwards instead of westwards1 as before4 ,the quadrantal error is therefore not eliminated1 but merely reversed in direction by adopting the simpler mounting for the compass4 ,,chapter ,,xii ,,the ,,elimination ,,of ,,the ,,quadrantal ,,error ,from our description of the cause of the quadrantal error it should be clear that it is of a variable erratic nature1 or at least that1 unlike the latitude and north steaming errors1 its magnitude cannot be forecast from a knowledge of the ship's speed1 course1 latitude1 or other factors4 ,its direction is determined by the direction of the ship's course1 but its amount is settled by the violence of the rolling and pitching1 and cannot therefore be calculated and tabulated in a practically useful way4 ,it follows1 therefore1 that to get rid of the upsetting influence of the quadrantal error we cannot resort to 8correcting0 the compass readings for it1 but must entirely eliminate it4 ,in the early 7#aiaj7 ,anschtz compass there were no means of eliminating the quadrantal error1 for the existence of the error1 or at least the importance to be attached to it1 was not at first recognised4 ,the compass1 we believe1 showed errors from this cause of as much as #bj deg4 to #dj deg4 ,as a result1 within a very short time the design was discarded1 and what amounted to an entirely different compass was substituted for it4 ,the #aiab ,anschtz compass will be briefly described later on1 but here it may be said that the quadrantal error in it is eliminated by adding two more gyro-wheels to the sensitive element4 ,the theory of this compass is not very easy to understand1 so that it will be best if we explain first how the quadrantal error is eliminated in the ,sperry and the ,brown compasses4 ,the early ,sperry1 like the early ,anschtz compass1 was open to the full action of the quadrantal error4 ,in the form now in use1 however1 it is prevented from taking effect by controlling automatically the position of the excentric pin connecting the weight or 8bail0 to the gyro casing4 ,in the ,sperry compass1 as we know1 the weight ,s 7,fig4 #ce7 is not hung from the inner supporting ring or its equivalent1 but from the follow-up or phantom ring1 its pendulous effect being communicated to the casing and thence to the spinning wheel through the excentric pin ,a4 ,the kicks of the weight at the out positions during the rolling of the ship are also transmitted through the pin to the casing and wheel inside it4 ,illustration3 ,,fig4 #ce4 ,sperry ,compass on ,n4,w4 course4 ,let us suppose1 as shown in ,fig4 #ce1 that at the port out position of the compass swing the excentric pin is east of the axis ,h ,j by the amount required just to bring it vertically below the centre of the spinning wheel4 ,then the southward kick of the weight when the vessel rolls on a north-west course is transmitted to the casing as a southward force applied at the point ,a4 ,this force1 acting about the axis ,e ,f as before1 produces precession in the direction ,k1 and1 as before1 we may resolve this movement into the components ,n ,m4 ,the force at ,a has1 however1 in addition1 a turning moment about the axis ,h ,j1 for it is applied to the casing at a point lying at a distance ,a ,d from this axis1 and not1 as in ,fig4 #cc1 at a point virtually on it4 ,this moment about ,h ,j would tend to make the end ,b of the axle turn in the direction of ,f1 and therefore results1 in accordance with the rule of the gyroscope1 in an actual movement about the axis ,e ,f1 the end ,b of the axle precessing downwards towards ,j1 that is1 in the direction ,r4 ,we may resolve the precession ,r into two components ,t ,u4 ,similarly1 at the starboard out position1 let us suppose that the excentric pin is this time westwards of the axis ,h ,j by exactly the amount required again to bring it vertically below the centre of the wheel4 ,as before1 in the case shown in ,fig4 #cc1 the northward kick of the weight acting about ,e ,f results in a precession ,l1 which may be resolved into the components ,p ,q4 ,the kick1 however1 as at the port out position1 also has a moment about the axis ,h ,j4 ,the kick is reversed in direction1 but the point of its application to the casing is now on the other side of ,h ,j4 ,consequently the kick tends to rotate the casing about ,h ,j--and therefore produces precession about ,e ,f--in the same direction as does the kick at the port out position4 ,the precession produced is represented by ,v1 and can be resolved into the components ,w ,x4 ,considering now the four pairs of component precessions1 we see that they cancel out in pairs4 ,thus ,n and ,q cancel1 ,u cancels ,x1 ,t wipes out ,m1 and ,w does the same to ,p4 ,the two components ,m ,p1 which were the cause of the quadrantal error1 are just balanced by the two additional components ,t ,w4 ,the axle therefore does not rise vertically1 but remains horizontal1 and as a result no quadrantal error can arise4 ,it will be seen that the ,sperry method of eliminating the quadrantal error requires the excentric pin to be movable relatively to the bail and casing4 ,to be more precise1 while the casing1 the bail1 the phantom ring1 and all the other parts of the compass may swing round the axle of the spinning wheel1 or the external gimbal axis coincident or parallel with the axle1 matters have to be arranged in such a way that when the ship rolls the excentric pin shall not partake of this motion1 but remain constantly in the vertical below the centre of the spinning wheel4 ,we have to remember1 however1 that the ,sperry method of damping the horizontal vibrations of the axle hangs upon the pin being displaced towards the east of the axis ,h ,j when the compass is in the even keel position4 ,the two requirements are met by so controlling the position of the pin relatively to the bail and casing that when the bail1 casing1 etc41 swing sideways under the influence of the rolling of the ship1 the pin is maintained at a fixed distance eastwards of the true vertical through the centre of the spinning wheel at all points in the oscillation of the bail1 casing1 etc4 ,illustration3 ,,fig4 #cf4 ,sperry ,ballistic ,gyro4 ,the stabilisation of the excentric pin in this manner is effected gyroscopically by means of the attachment shown in ,fig4 #cf4 ,this device consists of a small high-speed electrically driven gyroscope running inside a casing which is mounted rotatably on a vertical axis inside a stirrup frame4 ,this frame1 as shown in ,fig4 #cg1 is hung pendulum-wise on the north side of the main gyro casing1 the axis of its suspension being collinear with the axle of the main gyro-wheel4 ,the axle of the small gyro-wheel is thus aligned in the east and west direction4 ,the stirrup bracket is turned up horizontally below the casing of the small gyro1 and at its end is fitted with guides1 carrying a pair of rollers4 ,these rollers constitute the excentric pin and1 as shown in ,fig4 #cg1 engage within two curved channel-sectioned tracks attached one to the bail and one to the main gyro casing4 ,if when the bail1 casing1 etc41 swing under the influence of the ship's rolling the excentric pin should attempt to follow suit--either by reason of friction at the axis of suspension of the stirrup bracket or at the track rollers--the wheel and casing of the small gyro will start processing round the vertical axis inside the stirrup1 for the attempt is equivalent to an endeavour to tilt the small gyro-axle in an east and west vertical plane1 and therefore calls forth the usual gyroscopic reaction4 ,the precession of the small gyro on its vertical axis is made to react on the stirrup bracket by means of a spring connection between the bracket and the casing1 as shown in ,fig4 #cf4 ,the direction of spin of the small gyro-wheel is such that the force thus applied to the stirrup bracket opposes and just balances the frictional or other force trying to make it swing with the bail1 casing1 etc41 of the main gyro4 ,in this way the excentric pin as the vessel rolls is caused to maintain its original distance from the vertical line through the centre of the spinning wheel4 ,illustration3 ,,fig4 #cg4 ,stabilized ,excentric ,pin 7,sperry ,compass74 ,the suppression1 or rather the avoidance of the quadrantal error in the ,brown compass1 is achieved in a manner which is mechanically very distinct from that adopted for the same purpose in the ,sperry compass4 ,illustration3 ,,fig4 #ch4 ,diagram of ,brown ,compass4 ,an elementary diagram of the ,brown compass is given in ,fig4 #ch4 ,in this sketch ,a is the casing1 inside which the gyro-wheel mounted on the axle ,b 7,c7 rotates in the direction of the dotted arrow--,b1 as before1 representing the north-seeking end of the axle4 ,the casing is supported on an east and west horizontal axis ,e ,f inside a vertical ring1 this ring being journalled at ,h ,j inside a frame ,d1 the equivalent of the square frame in our simple model4 ,to obtain complete freedom for the gyro the frame ,d is mounted on an athwartship axis ,g ,k1 which is itself carried inside a ring journalled within the binnacle on an axis ,l parallel with the ship's longitudinal centre line4 ,the pendulum weight ,s is fixed to the lowest point of the frame ,d4 ,if the weight ,s and frame ,d be set swinging on the axis ,g ,k1 the swinging movement will1 of course1 be directly communicated through the journals ,h ,j to the vertical supporting ring1 but1 as the trunnions ,e ,f of the casing are really supported on knife edges1 the swinging movement of the frame ,d cannot be communicated from the vertical ring through the trunnions ,e ,f to the casing ,a1 and thence to the wheel4 ,yet it is essential that the weight ,s should be able to act pendulum-wise on the casing and wheel1 for otherwise1 as we know1 the system would be without directive force4 ,illustration3 ,,fig4 #ci4 ,oil ,control ,bottles 7,brown ,compass74 ,the connection between the weight and the casing is not a mechanical one1 but is effected by making use of the air blast created by the fan-like action of the wheel inside the casing4 ,as we have explained in connection with the damping system adopted in this compass1 the trunnion ,f is hollow1 and delivers the air blast through a nozzle ,m1 fixed relatively to the vertical supporting ring1 into a divided box ,n4 ,from this box pipes are led to two oil bottles--one of which is shown at ,p--fixed to the casing on the east side of the axle1 one bottle being on the north face of the casing and the other on the south4 ,the pressure of the air blast acting unequally upon the oil in these two bottles when the casing tilts about the axis ,e ,f results1 as we have already explained1 in any horizontal oscillation of the sensitive element about the axis ,h ,j being damped4 ,in a similar way two bottles ,q ,r half-filled with oil are fixed to the north and south faces of the casing on the west side of the axle of the spinning wheel4 ,these bottles are also connected to the box ,n1 but the connecting pipes are crossed so as to join each bottle ,q ,r to the division of the box remote from it1 and not to the adjacent division1 as in the case of the damping bottles ,p4 ,when1 therefore1 the weight ,s and frame ,d are swung slowly pendulum-wise on the axis ,g ,k1 as shown in the first view in ,fig4 #ci1 the nozzle ,m1 being fixed to the vertical ring1 delivers more air into one division of the box ,n than the other1 and therefore a greater air pressure is exerted inside one bottle than inside the other4 ,consequently oil flows through the connecting pipe ,t from the former bottle to the latter until the columns of oil are sufficiently unequal to balance the difference of pressure of the air inside the bottles4 ,it will be noticed that the crossing of the pipes connecting the bottles with the box ,n results in the oil being accumulated in that bottle1 which lies away from the side to which the weight ,s has been swung4 ,hence1 although there is no mechanical connection between the weight and the casing of the spinning wheel1 the effect when the weight ,s is swung slowly is thus substantially the same as it would be if there were1 for the weight of the extra oil forced into the bottle ,r exerts a turning moment on the casing about the axis ,e ,f1 and tends therefore to make the casing follow the deflection of the weight ,s and frame ,d4 ,similarly1 should the axle ,b ,c dip1 as shown in the second view in ,fig4 #ci1 extra oil will accumulate in the bottle which has been elevated by the dipping1 and as a result a restoring moment about the axis ,e ,f will be applied to the casing1 just as it would be if the weight ,s had been directly connected to the casing and had been deflected by the dipping movement4 ,the second view in ,fig4 #ci illustrates the generation of the directive force in the ,brown compass4 ,let us suppose that the compass is at the equator1 and that by some agency the axle is turned so that its end ,b points due west4 ,the rotation of the earth will cause the axle to dip slowly into some such position as that shown4 ,during this slow dipping movement oil will flow slowly from the bottle ,q into the bottle ,r4 ,the unbalanced weight of oil will apply a turning moment to the sensitive element about the axis ,e ,f and will therefore produce actual motion about the axis ,h ,j1 which will precess the end ,b of the axle towards the north4 ,any tendency for the axle to vibrate about the north and south line as a result of the momentum acquired by the sensitive element while coming up from the west will be damped by the air blast acting upon the oil in the two other bottles ,p fixed1 as shown in ,fig4 #ch1 on the east side of the axle4 ,illustration3 ,,fig4 #dj4 ,brown ,compass on ,west ,course4 ,the point we require here to notice especially about this action is that1 as the tilting of the axle produced by the rotation of the earth takes place very slowly--it cannot exceed the speed of rotation of the earth on its polar axis1 namely1 #j4jjjg of a revolution per minute--the flow of oil from the bottle ,q to the bottle ,r takes place very slowly also4 ,the oil therefore acquires practically no momentum1 and rises in the bottle ,r in strict accordance with the tilt acquired by the axle4 ,if the tilting motion be stopped or reversed1 the oil would remain in the bottle ,r at just the level it had reached1 or would immediately begin to flow back again1 for its momentum being negligible1 it has not sufficient kinetic energy to rise higher in the bottle ,r after the upward motion of the bottle has ceased4 ,on the other hand1 when the ship rolls1 say1 on a due west course1 as represented in ,fig4 #dj1 the outer gimbal ring supporting the athwartship axis ,g ,k1 the frame ,d carrying the weight ,s1 and the vertical supporting ring will acquire an oscillation about the fore and aft axis ,l in tune with the rolls of the ship4 ,the air blast will thus be directed into the two divisions of the box alternately1 and therefore oil will flow from one bottle to the other4 ,it is to be noticed1 however1 that the rolling motion of the ship induces in this manner a flow of oil from one bottle to the other at a very much greater rate than does the tilting action of the earth's rotation dealt with above4 ,a ship rolling through #de deg41 out to out1 in a complete period of #aj seconds rotates about its rolling centre with an average velocity equivalent to #a4e revolutions per minute--or over #bjjj times as fast as the speed of rotation of the earth on its polar axis--and at the mid point of its roll it will move with an actual velocity of about twice the average4 ,the momentum acquired by the oil in flowing from bottle to bottle is therefore in this case not negligible4 ,in fact1 when the ship reaches one of its out positions and starts to return1 the oil does not immediately begin to flow back into the other bottle1 but is carried by its kinetic energy to a still higher level in the bottle in which it has been rising4 ,as a result1 the oscillation of the oil between the two bottles lags behind the oscillation of the pendulum weight ,s1 and therefore that of the ship itself4 ,the lag acquired is such that the oil is just level in the two bottles when the ship is at either of its out positions1 and when the ship is at the mid point1 or even keel position of its roll1 although the air blast is for the moment evenly distributed between the two compartments of the divided box1 the oil is standing at its maximum level in the bottle on that side of the wheel from which the ship is recovering herself4 ,the action of the oil in the bottles during a rapid oscillation of the compass system is1 in fact1 quite analogous to that of the water in the ,frahm system of anti-rolling tanks4 ,it will thus be seen that the net effect of transmitting the 8kicks0 derived during rolling from the pendulous weight of the wheel through the ,brown oil bottle arrangement is simply to delay the application of the kicks to the gyro-wheel by a constant amount1 namely1 by the time taken for the ship to roll from either out position to the mid position or a quarter of a complete period4 ,thus as the ship sailing due west rolls through the mid position from starboard to port the compass system experiences the turning moment about the axis ,e ,f1 which with a rigidly fixed pendulum it would receive at the starboard out position from the northwards kick of the weight4 ,similarly1 the equivalent of the southwards kick of the weight at the port out position is felt by the wheel when the compass is passing through the even keel condition on the subsequent roll from port to starboard4 ,it is clear that with the ship sailing due west--or east--this delaying of the kick does not affect the result established previously for a rigidly connected pendulous weight4 ,any tendency for the axle to precess towards the west when the compass is passing through the even keel position in one direction is completely annulled by the tendency to precess towards the east at the succeeding passage through the even keel position in the opposite direction4 ,on quadrantal courses1 however1 the delay in the application of the kick is most important1 for when the ship rolls it results in the elimination of the quadrantal error4 ,that it does so can easily be understood by reference to ,fig4 #cc4 ,the kicks1 being received when the compass is passing through the even keel position1 and not at the out positions1 precess the wheel about the axis ,h ,j at a time when this axis is truly vertical1 and not when it is inclined4 ,the precessional movements have therefore no vertical components ,m ,p4 ,they are represented completely by the horizontal components ,n ,q4 ,the axle thus does not depart from the horizontal plane1 and any movements in this plane arising from the tendency to precess in the directions ,n ,q cancel each other at successive passages of the compass through the mid position4 ,the early form of ,anschtz compass was followed by the #aiab pattern in which the quadrantal error was successfully eliminated4 ,an example of the #aiaj form was obtained by ,messrs4 ,elliott ,brothers1 of ,lewisham1 from ,anschtz and ,co41 of ,kiel1 and was1 we believe1 fitted on board ,h4,m4,s4 ,new ,zealand4 ,its defects becoming apparent1 its manufacture in this country was not proceeded with1 but upon the appearance of the improved type in #aiab1 ,messrs4 ,elliott took up its construction and supplied several to the ,admiralty4 ,the ,sperry compass1 however1 secured the preference in the ,british ,navy1 and with the outbreak of the war ,messrs4 ,elliott ceased practically to make ,anschtz compasses4 ,on the other hand1 the ,german ,navy continued to use the #aiab type of ,anschtz compass1 with very little alteration or addition1 right throughout the war4 ,in view of the fact that every ,german submarine was fitted with a compass of this form1 and bearing in mind the high degree of excellence attained in the navigation of the enemy's underwater craft1 there can be no doubt that the modern ,anschtz compass is a very satisfactory device4 ,in ,fig4 #da we give a purely diagrammatic representation of the compass4 ,the outer square frame may1 as usual1 be regarded as mounted within external gimbal rings1 providing a fore-and-aft axis and an athwartship axis4 ,the square frame contains a vertical ring free to turn about the axis ,h ,j and itself containing a horizontal ring mounted on an east and west axis ,f ,e4 ,the pendulum weight ,s is attached1 as usual1 to the inner horizontal ring4 ,the essential difference between the #aiab and the #aiaj forms of ,anschtz compass lies in the fact that1 as shown in our diagram1 the inner ring1 or its equivalent1 does not surround a single gyro-wheel1 but has attached to it the casings of three distinct gyros4 ,illustration3 ,,fig4 #da4 ,diagram of ,anschtz 7#aiab7 ,compass4 ,as shown in the first plan--in ,fig4 #db--the three gyros are situated at the corners of an equilateral triangle4 ,one gyro ,k is placed at the south end of the meridional diameter of the horizontal ring1 with its axle pointing towards the centre of the ring4 ,the two other gyros ,l ,m are placed at #fj deg4 east and #fj deg4 west of north1 with their axes pointing towards the centre of the gyro ,k--not towards the centre of the ring4 ,the wheels of all three gyros rotate anti-clockwise1 as seen from the north1 looking south--that is1 they rotate in the same direction as do the wheels of all single-gyro compasses as seen from the same standpoint4 ,illustration3 ,,fig4 #db4 ,plans of ,gyros4 ,if we neglect the two gyros ,l ,m for the moment1 imagining them to be replaced by simple deadweights to counterbalance the weight of the gyro ,k1 it will be seen that this system differs from that of the early ,anschtz compass only in the fact that the gyro-wheel has been reduced in size1 and has been displaced from the centre of the horizontal ring to the southern edge4 ,the displacement of the wheel in this manner in no way affects the essential working of the compass4 ,in a single-gyro compass we might with some advantage similarly displace the spinning wheel1 for by so doing and by counterbalancing the displaced weight we should1 at least partially1 get rid of the concentration of mass about the east-west axis1 which1 as we shall see1 introduces1 if not corrected1 an additional source of error into the compass readings when the ship rolls on quadrantal courses4 ,the reduction in the size of the gyro-wheel would1 of course1 reduce the magnitude of the directive force4 ,in the #aiab design the wheels of all three gyros run at #bj1jjj revolutions per minute--that is1 at the same speed as the wheel in the #aiaj design4 ,but each wheel is only #e in4 in diameter instead of #f in41 and weighs1 with its axle1 #e lb4 #b oz41 or only half as much as the wheel in the earlier form4 ,the gyro ,k considered alone would therefore supply a directive force of but half the former amount4#f ,it follows1 obviously1 that if a second gyro of the same speed and size as that at ,k be fixed in accurate alignment at the north end of the meridional diameter of the horizontal ring the directive force at any angle of horizontal deflection away from the north will be doubled1 that is1 will be made equal to that developed in the #aiaj design4 ,such a compass might be constructed1 but it would exhibit the quadrantal error--or at least the inertia force portion of that error--just as badly as did the early ,anschtz4 ,the essence of the #aiab design lies in the fact that not one but two gyros are attached to the north side of the horizontal ring1 and in the additional fact that the axles of these two gyros are not parallel with but are inclined to the axle of the south gyro ,k4 ,when the sensitive element of this compass is in the north resting position1 the axle of the gyro ,k is aligned with the meridian and therefore the rotation of the earth--the compass being supposed at the equator--merely moves the axle parallel with itself4 ,on the other hand1 with the sensitive element in the north resting position1 the axles of the gyros ,l ,m are inclined to the meridian at #cj deg4 ,the gyro ,l is virtually in the condition of a single-gyro compass1 with the north end of its axle partially turned towards the east4 ,under this condition1 as we know1 the rotation of the earth will tend to make the north end ,b of the axle rise above the horizontal plane4 ,conversely1 the gyro ,m is in the condition of a single-gyro compass1 with the north end of its axle partially turned towards the west4 ,the earth's rotation in this case tends to make the north end of the axle dip below the horizontal plane4 ,thus1 in the #aiab ,anschtz compass1 when the sensitive element is in the north resting position1 the gyro ,k under the rotation of the earth is without effect on the pendulous weight ,s1 the gyro ,l is striving to swing it towards the north1 and the gyro ,m is trying with an equal effort to swing it towards the south4 ,the weight1 therefore1 remains in the plumb line and applies no turning moment to any of the gyros4 ,as there is no turning moment1 there is no precessional tendency4 ,the sensitive element remains directed towards the north4 ,this alignment1 as in a single-gyro system1 is the true resting position1 and under it no directive force is applied to the sensitive element4 ,to study how the three gyros act together to restore the sensitive element to the north resting position should it be deflected therefrom1 let us suppose that the deflection suffered is one of #cj deg4 towards the east4 ,as shown in the second plan in ,fig4 #db1 the effect of such a deflection is to place the three gyros ,k ,l ,m in the condition respectively of a single-gyro system when the axle is 7k7 deflected #cj deg4 to the east1 7l7 deflected #fj deg4 to the east1 and 7m7 aligned on the north4 ,at this deflection1 then1 the gyro ,m contributes no restoring force4 ,the directive force contributed by the gyro ,k is half that contributed by the single gyro of the #aiaj design when the deflection is #cj deg41 for the mass--or moment of inertia--of the wheel is half the earlier value4 ,the directive force contributed by the gyro ,l is something greater than that contributed by the gyro ,k1 for the virtual deflection eastwards is #fj deg4 instead of #cj deg41 and1 as we know1 the directive force increases with the deflection4 ,thus1 the three gyros taken together supply a directive force1 when the sensitive element is deflected through #cj deg41 which is somewhat greater than that supplied by the single double-sized wheel of the #aiaj design4#g ,this result is a general one4 ,whatever the deflection may be1 the directive force supplied by the three-gyro compass is always about one-third greater than the directive force of the #aiaj design at the same deflection4 ,this increased force is developed even if the deflection be less than #cj deg4 ,in such cases the gyro ,m will1 of course1 supply a force1 a non-restoring force1 to the sensitive element4 ,only when the deflection exceeds #cj deg4 eastwards does the gyro ,m assist the gyros ,k and ,l to restore the sensitive element to the north resting position4 ,if the deflection be to the west1 however1 the gyro ,m is the chief assistant of the gyro ,k1 a laggard's part being played by the gyro ,l until #cj deg4 of westerly deflection is reached4 ,the manner in which this three-gyro system avoids the quadrantal error can now be discussed4 ,the quadrantal error1 it may be recalled1 arises when the ship rolls on an intercardinal course1 and is primarily caused by the fact that the whole compass system can swing on its external gimbal axis in tune with the rolls of the ship4 ,if we could so arrange matters that during the roll of the ship from side to side the compass system would swing on the external gimbals1 no more and no less than the amount required just to keep the axis ,h ,j truly vertical at all points of the roll1 then the north and south 8kicks0 received on the weight ,s at the out positions of the roll would cause the sensitive element to precess first one way1 then the other1 but always in a horizontal plane4 ,there would be no vertical component in the precession1 the cumulative effect of which1 as we have seen1 produces the quadrantal error4 ,in the ,brown compass1 the quadrantal error is eliminated by delaying the effect of the 8kicks0 on the weight ,s until the axis ,h ,j is truly vertical--that is to say1 the 8kicks0 on the weight are not transmitted to the spinning wheel until the compass system is passing through the even keel position4 ,in the ,sperry compass the weight ,s is virtually shifted back and forth from east to west of the axis ,h ,j in tune with the rolling of the ship in such a way as to introduce a second component of vertical precession1 which in amount and direction is just sufficient to nullify the vertical component causing the quadrantal error4 ,in the #aiab ,anschtz compass1 the object aimed at is the maintenance of the axis ,h ,j truly vertical at all times during the rolling condition4 ,this object is achieved1 as we shall now show1 by setting the axles of the gyros ,l ,m--,fig4 #da--;inclined to the axle of the gyro ,k4 ,in a single-gyro compass1 or in the #aiab ,anschtz compass with the gyros ,l ,m suppressed1 the system is very stiff against vibrations on the east-west axis ,e ,f1 but is quite easily set vibrating about the north-south axis provided by the external gimbal mounting4 ,the reason is1 of course1 that any vibration about the axis ,e ,f is met by the resistance of the gyro-wheel1 for the vibration causes the axle to alter its direction1 whereas no gyroscopic resistance is called into play by the vibration on the north and south axis1 for this axis is coincident or parallel with the gyro-axle4 ,thus1 in the #aiaj ,anschtz compass the period of vibration about the east-west axis was something like #gj min41 whereas the period about the north and south axis was only one or two seconds4 ,the smallness of the latter period readily resulted in the compass system getting into a swing in tune with the period of rolling of the ship--from five to twelve seconds or so4 ,were the axles of the gyros ,l ,m in the #aiab form set parallel with that of the gyro ,k1 there would still be no gyroscopic resistance exerted against vibration on the north-south axis4 ,as it is1 the inclination of the axles of these two gyros1 taken together1 has virtually the same effect as would be obtained by the addition to the sensitive element of a separate single gyro1 with its axle aligned at right angles to the axle of the gyro ,k4 ,the gyroscopic stiffness against vibration about the east-west axis ,e ,f is hardly affected by the inclination of the gyros ,l ,m at the angle chosen by the designers4 ,the resistance1 instead of being that contributed by three gyros1 is equivalent to that supplied by about #b gyros4 ,the period of vibration on this axis ,e ,f is arranged to be the standard #he min4 ,about the north and south axis we have virtually the resistance of one gyro4 ,the period of vibration about this axis1 instead of being only one or two seconds1 is about #hj sec4 ,with this lengthened period of vibration there is practically no chance of the compass system getting into a swing in tune with the rolling of the ship1 for every #b to #f seconds the vibrating influence is reversed in direction1 and in this interval of time the system cannot acquire any substantial degree of swinging motion4 ,thus1 as the compass1 following the rolls of the ship1 moves from side to side1 it suffers a pure translational movement4 ,the axis ,h ,j remains truly vertical at all instants during the roll1 and therefore the north and south kicks of the pendulous weight at the out positions of the roll precess the gyro-axle about ,h ,j in purely a horizontal plane1 the precessional tendency at one out position cancelling that at the other4 ,there is no vertical component in the precession1 and therefore there can be no quadrantal error4 ,it is of interest and of some amusement to note that during the war the ,germans applied a fourth gyro to the ,anschtz compass1 and that this additional gyro was carefully removed from every compass before they surrendered their submarines to us4 ,the deceit was1 however1 of no avail1 for the application of the fourth gyro was known to us long before the war ended1 a complete compass so fitted having been recovered from a sunken submarine and repaired and carefully studied4 ,the fourth gyro was applied to the external gimbal rings4 ,these rings1 when the submarine rolled1 were found to acquire at times a violent oscillation of their own1 for1 of course1 they received no stabilisation from the gyroscopic elements of the compass4 ,on board submarines the violence with which the rings vibrated would occasionally threaten to wreck the compass4 ,by adding a gyroscope to the gimbal rings1 the period of vibration of the rings was lengthened to about #af sec41 so that little or no opportunity to swing was left to them4 ,with the exception of this addition1 the ,anschtz #aiab compass was used by the ,germans throughout the war practically without alteration4 ,,chapter ,,xiii ,,centrifugal ,,forces ,,during ,,quadrantal ,,rolling ,in addition to the 8kicks0 of the pendulous weight1 which during quadrantal rolling1 as we have seen1 react gyroscopically upon the compass and tend to make the axle deviate from the north1 there is a second influence at work on the compass1 which1 when the vessel rolls on an intercardinal course1 likewise tends to deviate the axle4 ,the two deviations are always in the same direction1 so that we cannot arrange the one to reduce or eliminate the other4 ,the second deviation arises from the centrifugal forces developed in the compass parts during quadrantal rolling4 ,these forces1 like the kicks of the pendulous weight1 react gyroscopically upon the spinning wheel1 and if not checked will deviate the axle away from the north4 ,illustration3 ,,fig4 #dc4 ,centrifugal ,forces on a ,pendulum4 ,let us consider the somewhat unusual type of pendulum shown in ,fig4 #dc1 a pendulum the rod ,a of which is fixed rigidly to the bar ,b carried on knife edges1 and of which the 8bob0 is in the form of a cylinder suspended at its mid point from the rod ,a in such a way as to permit the bob to turn horizontally on the bearing at ,c1 substantially without friction4 ,in the first instance1 let the pendulum be set swinging about the knife edges with the cylindrical bob set parallel with the axis ,b4 ,each portion of the bob is thus caused to swing about that point on the axis ,b which is vertically over the portion when the bob is at rest4 ,the centrifugal force developed on each portion is at all positions of the swing directed radially from such point4 ,the centrifugal force on each portion is a maximum when the bob is passing through the mid position1 and falls1 with the velocity1 to zero at each of the out positions4 ,at the mid position the centrifugal force on the two extreme portions ,l ,m of the bob may be represented by ,d ,e4 ,these two forces are equal and are directed vertically downwards4 ,on all other similar portions the centrifugal force at the same instant is similar in magnitude and direction4 ,the net effect of the centrifugal forces on all the portions is thus simply a tendency to bend the bob ends downwards about the central line at ,c4 ,they have clearly no tendency either at the mid position of the swing or at any other to turn the bob round the axis ,c at the foot of the rod4 ,let us now set the pendulum swinging with the bob arranged at right angles to the knife-edge axis1 as shown in the second view of ,fig4 #dc4 ,each and every portion of the bob is now swinging about the one point ,f on the axis4 ,at any instant in the swing the centrifugal force on any portion is directed along the line joining the point ,f to the centre of that portion4 ,taking the two extreme end portions ,l ,m as the bob passes through the mid position1 the centrifugal forces are as indicated at ,g ,h4 ,they are not now vertical nor parallel with each other1 but1 again1 they have no tendency to turn the bob horizontally about the axis at ,c4 ,resolving them into vertical and horizontal components1 we see that the net effect of all the centrifugal forces is to try to bend the ends of the bob downwards--the effect of the vertical components--combined with the application to the bob of a horizontal stretching force--the effect of the horizontal components4 ,let us1 however1 rotate the bob about the axis at ,c through something less than #ij deg4 from the first position--say1 through #de deg41 as indicated in the third sketch4 ,consider again the two extreme end portions as the bob is passing through the mid position of its swing4 ,let ,j ,k be a line through the centre of the bob parallel with the knife-edge axis4 ,from the centres of the end portions1 let lines ,l ,n and ,m ,p be drawn at right angles to ,j ,k4 ,from ,n and ,p draw lines upwards parallel with the rod ,a to meet the knife-edge axis at ,q and ,r4 ,then ,q and ,r are the points about which the two extreme portions of the bob are swinging4 ,the centrifugal forces on these portions are directed along the lines ,q ,l and ,r ,m1 and are indicated at ,s and ,t4 ,these forces may1 as shown1 be resolved into vertical components--parallel with ,q ,n and ,r ,p--and horizontal components--parallel with ,n ,l and ,p ,m4 ,treating the other portions of the bob similarly1 it will be seen that while the vertical components of the centrifugal forces1 as before1 merely tend to bend the ends of the bob downwards1 the horizontal components are no longer parallel with the length of the bob1 as in the second sketch4 ,they are inclined to the bob and clearly apply to it a moment which will turn it about the axis at ,c in the direction of the arrow ,u until the bob reaches the position shown in the second sketch4 ,it is obvious that any rotation of the bob on the axis at ,c away from the position shown in the first sketch1 even a slight one1 will call into action horizontal components of centrifugal force after the manner shown in the third sketch4 ,the equilibrium of the bob in the position shown in the first sketch is therefore unstable4 ,with the bob in the second position a movement of the end ,l to the left will1 as we know1 call into action horizontal components of centrifugal force1 tending to turn the bob about ,c in the direction of the arrow ,u4 ,if the end ,l is moved to the right the horizontal components1 as can easily be verified by repeating the argument1 will tend to turn the bob about ,c in the direction opposed to the arrow ,u4 ,thus deflection to either side of the second position calls forth forces which lead the bob to recover that position4 ,we conclude1 therefore1 that the second position is one of stable equilibrium4 ,it may further be shown by repeating the argument that if the bob is rotated about ,c from the first position in such a way as to send the end ,l to the rear the horizontal components called into action tend to set the bob into the second position with the end ,l pointing backwards1 and that when the bob is so aligned relatively to the knife-edge axis the equilibrium is again stable4 ,thus1 altogether1 if the bob is parallel with the knife-edge axis1 the equilibrium is unstable4 ,if it is at right angles the equilibrium is stable4 ,if it is set into any intermediate position1 forces are called into play which endeavour to align it at right angles to the knife-edge axis4 ,let a second cylindrical bob of the same size and weight be attached to the first at right angles to it and in the same plane4 ,then in all positions of the bobs relatively to the knife-edge axis the equilibrium will be stable1 for at any setting of the bobs the rotary force about ,c developed by the one bob will be equal and opposed in direction to that developed by the other bob4 ,the above discussion covers a principle of considerable general importance in mathematical physics4 ,briefly put1 it shows us that a body suspended pendulum-wise in the manner sketched will develop a tendency when swung to set its longer axis parallel with the plane of the swing1 and that to avoid this tendency in a swinging body mounted in the manner we have indicated1 the mass of the body must be distributed symmetrically and equally about the pendulum rod1 so that there shall be no 8longer axis40#h ,so far as the gyro-compass is concerned1 the circumstances described necessitate--at least in the ,sperry and ,brown compasses--the addition of compensator weights to represent the second cylindrical bob to which we have referred above4 ,the masses composing the gyro-compass are not disposed symmetrically and equally about the vertical axis--the axis which we have throughout called ,h ,j4 ,as seen in plan1 the masses are concentrated towards the east and west plane containing the spinning wheel1 and are deficient towards the north and south4 ,by virtue of the presence of the pendulous weight1 or its equivalent1 below the spinning wheel1 the whole mass of the compass is capable of swinging pendulum-wise1 the axis of this swing being one or other of the axes of the external gimbal mounting4 ,further1 the swinging mass is free--if the wheel is not spinning--to turn about the vertical axis which we have called ,h ,j4 ,reference to ,fig4 #ca will make these statements quite clear4 ,thus the compass and its system of mounting inevitably reproduce the essential features of the pendulum shown in ,fig4 #dc4 ,taking the second sketch in that engraving1 the bob represents the masses of the compass concentrated in the east and west plane4 ,the gyro-axle being at right angles to this plane is parallel with the knife-edge axis ,b4 ,this axis ,b may be regarded as representing the longitudinal axis of the external gimbal mounting1 so that in the conditions imagined the ship is supposed to be sailing due north or south4 ,it is clear that when the ship rolls on either of these courses1 the fact that the mass of the compass is concentrated in the east and west plane will not tend to deviate the axle1 for the 8pendulum bob0 is in its position of stable equilibrium4 ,the first sketch similarly represents the condition of matters existing when the ship rolls on an east or west course1 the knife-edge axis ,b again being identified with the longitudinal axis of the external gimbal mounting4 ,the pendulum bob is now in its position of unstable equilibrium1 so that if the mass distribution of the compass is not corrected by the addition of compensator weights1 a tendency for the axle to deviate away from the north may occur4 ,if we identify the knife-edge axis ,b with the athwartship axis of the external gimbal mounting1 the second sketch in ,fig4 #dc illustrates the condition of matters existing when the ship pitches on an east or west course1 while the first will represent pitching on a due north or south course4 ,on quadrantal courses1 failure to correct the inequality of the mass distribution1 if that distribution is not uniform1 will inevitably result in the axle deviating when the ship rolls4 ,thus1 the third view in ,fig4 #dc represents the conditions prevailing when the ship rolls on a due north-west--or south-east--course4 ,looking at the plan1 the line ,j ,k represents the direction of the longitudinal axis of the external gimbal mounting and therefore the direction of the ship's course1 while a line at right angles to the pendulum bob would represent the gyro-axle1 and therefore the north and south direction4 ,when the ship rolls the compass masses oscillate pendulum-wise about the external longitudinal gimbal axis--represented by the bar ,b--but the directive force of the compass tends to preserve the general plane of the oscillating masses inclined at #de deg4 to the direction of the bar ,b4 ,as a result1 the horizontal components of the centrifugal forces developed in the compass masses apply to the wheel1 casing1 etc41 a turning moment in the direction of the arrow ,u4 ,this moment1 it will be seen1 tends to turn the sensitive element in the same direction as that in which the 8kicks0 of the weight at the out positions of the swing endeavour to deviate the axle on the same course4 ,the centrifugal moment would1 however1 cause an actual deviation in the direction ,u in a direct manner only if the gyro-wheel were not spinning4 ,as it is1 the centrifugal moment causes the north end of the gyro-axle to precess upwards and therefore leads to a deflection of the pendulous weight of the north4 ,this deflection throws a turning moment on to the wheel about the horizontal axis--the axis we have called ,e ,f throughout--and1 finally1 this turning moment precesses the axle horizontally in the direction of the arrow ,u4 ,thus1 with the wheel spinning1 the centrifugal moment actually does produce rotation in the direction ,u1 but its action is not direct4 ,as we have said1 the tendency of the axle to deviate during rolling on quadrantal courses as a result of the action of centrifugal forces1 is corrected in those compasses requiring it by the addition to the compass masses of compensator weights on the north and south sides of the sensitive element4 ,these weights are adjusted for mass and position in such a way as to make the mass distribution of the sensitive element substantially the same in the north and south direction as in the east and west4 ,in this way the centrifugal forces tending to cause deviation are neutralised on all courses1 both during rolling and pitching4 ,,chapter ,,xiv ,,the ,,ansch,,tz 7#aiaj7 ,,compass ,some few words of description fall to be made regarding certain mechanical and electrical features of the principal types of gyro-compass--features1 that is1 which1 not being primarily connected with the gyroscopic behaviour of the compass1 have not so far been mentioned in our discussion of the general theory of the device4 ,a sectional diagrammatic view of the early form of ,anschtz compass is given in ,fig4 #dd4 ,this compass1 although it is now obsolete by reason of the fact that no provision was made in its design for the quadrantal error1 is still1 we think1 of interest from other than the historical point of view4 ,illustration3 ,,fig4 #dd4 ,the ,anschtz 7#aiaj7 ,compass4 ,the gyro-wheel is shown at ,a4 ,the axle is carried in bearings inside the casing ,b4 ,this casing is attached to a tubular stalk ,c1 which is fixed to a head ,d carrying the compass card ,r4 ,this system of parts as a whole is floated in mercury ,q contained inside a circular steel bowl ,k1 the flotation being secured by means of a hollow steel ring ,s joined to the head ,d by a dome-shaped member ,e1 which is perforated all over for the sake of lightness4 ,as the float ,s is completely immersed in the mercury1 the exact level to which it will sink can be delicately settled by controlling the form of the bore inside the head ,d4 ,the central position of the float inside the bowl is controlled by means of a steel stem ,t fixed at its upper end to the centre of the glass ,g covering the card and pointed at its lower end to dip into a cup containing mercury which is fixed centrally inside the stalk ,c4 ,the stem ,t is insulated1 and is enclosed within a tube1 the upper end of which is also attached to the glass ,g and the lower end of which is splayed out to dip into a second mercury cup surrounding but insulated from the first4 ,through this tube and its mercury cup one phase of a three-phase current is led to the motor driving the gyro-wheel from the terminal ,f4 ,the second phase reaches the motor from the terminal ,h by way of the stem ,t1 while the third phase is transmitted through the bowl1 the mercury ,q1 and the float4 ,the bowl is provided with knife-edge bearings at ,l1 whereby the whole system shown is mounted within two gimbal rings providing the external longitudinal and athwartship axes4 ,the outer gimbal ring is supported inside the binnacle by means of springs1 the attachments of these springs1 as well as the gimbal supports1 being insulated on account of the method adopted for transmitting the third phase of the current to the gyro-motor4 ,the gyro-axle runs on ball bearings4 ,the driving motor comprises a stator carrying the windings and fixed inside the casing ,b and a rotor forming a rigid part of the gyro-wheel itself4 ,it will be seen that this design of compass does not explicitly reproduce the horizontal axis ,e ,f1 nor the pendulous weight ,s of our simple model4 ,the float is1 however1 free to tilt within the bowl in any direction1 so that the system of suspension is practically equivalent to the provision of an infinite number of horizontal axes ,e ,f4 ,as regards the apparent absence of the pendulous weight1 it is to be noted that the centre of gravity of the floating system is below its metacentre1 and therefore that the pendulum effect is exactly reproduced4 ,the damping details are not shown in the engraving1 but they are substantially as we have described them previously4 ,in view of the high speed1 #bj1jjj revolutions per minute1 at which the gyro-wheel runs1 giving a stress in the rim of about #aj tons per square inch and a peripheral velocity of #cdj miles an hour1 it is of interest to note that during a test to destruction the wheel did not fail until it was being driven at a speed involving the supply to it of five times the normal driving power4 ,a special motor generator had to be built to enable this test to be carried out4 ,even the normal motor driving the gyro-wheel had to be specially designed1 for no motor capable of running at #bj1jjj revolutions was at the time commercially available1 while the small space within which it had to be accommodated rendered the question of temperature rise a very serious one4 ,it was found1 it may be added1 that in designing the motor the usually accepted magnetic constants for the iron in the motor did not hold good with a periodicity as high as #ccc cycles per second4 ,the flotation of the sensitive element in mercury is a simple and convenient method of obtaining a practically frictionless support for the vertical axis--the axis ,h ,j of our elementary model4 ,the absence of friction arises from the facts that the drag of the mercury on the parts of the sensitive element with which it is in contact is proportional to the velocity with which the parts move through it1 and that this velocity is always extremely slow4 ,if the mercury is freshly distilled1 there is1 it would appear1 substantial absence of frictional drag4 ,but in course of time dust1 oil1 etc41 may collect on the surface of the mercury1 and may introduce a drag on the sensitive element of sufficient amount to interfere with the accuracy of the compass readings4 ,the suspension of the vertical axis by flotation has1 however1 the advantage that no external gimbal rings need be provided to give the sensitive element three degrees of freedom in all configurations4 ,the sensitive element1 as we have said1 is virtually provided with an infinite number of horizontal axes1 so that within the bowl itself there is always a longitudinal and an athwartship axis about which the sensitive element may turn4 ,the external gimbal axes provided in this compass permit the bowl to remain horizontal when the ship rolls or pitches2 they are not essential gyroscopically4 ,a similar remark applies to the later form of ,anschtz compass in which the sensitive element is also floated in mercury4 ,,chapter ,,xv ,,the ,,sperry ,,compass ,a general view of the ,sperry compass as removed from the binnacle is given in ,fig4 #de4 ,this view depicts the compass as it would be seen looking forward on a vessel steaming towards the south-west4 ,at ,a is shown the compensator weight on the north side of the sensitive element1 and at ,b the ballistic gyro is to be seen4 ,the ring ,c is the inner of the two external gimbal rings1 and carries on an athwartship axis ,d the spider ,e1 from which the sensitive element is suspended4 ,at ,f may be seen one of the bearings whereon this ring is swung on a longitudinal axis within the outer gimbal ring4 ,the outer ring1 in turn1 is hung on springs attached to the inside of the binnacle4 ,the speed and latitude corrector dials are to be seen directly above the bearing ,f1 while behind them the inclined cosine ring or course corrector ring is visible4 ,the phantom ring is shown at ,g4 ,this ring in section is channel-shaped1 and in the view hides the vertical ring within which the casing of the gyro-wheel is journalled on the horizontal east and west axis4 ,it is to be carefully noted that the lugs ,h1 by which the compensator weights are attached to the sensitive element1 are not fixed to the phantom ring1 but pass through easy-fitting holes therein1 and are directly secured to the vertical ring inside4 ,the bail weight lies within the phantom and vertical rings1 and can be seen at ,j1 the tracks for the excentric pin1 one on the bail and the other on the wheel casing1 being shown at ,k4 ,the stirrups ,l are fixed to the ends of the bail1 and carry the pins whereon the bail is swung from the phantom ring4 ,the bail latitude corrector dial1 whereby the bail is allowed to assume the tilt appropriate to each latitude without disturbing the horizontality of the gyro-axle1 is visible behind the stirrup4 ,the horizontality of the axle is indicated on a level attached at this point to two brackets springing from the wheel casing4 ,illustration3 ,,fig4 #de4 ,the ,sperry ,compass ,removed from ,binnacle4 ,illustration3 ,,fig4 #df4 ,the ,sperry ,compass4 ,of the non-gyroscopic mechanical details of this compass1 chief interest probably centres in the method of providing the vertical axis about which the sensitive element may turn relatively to the spider ,e in a substantially frictionless manner4 ,in ,fig4 #df we give a diagram of the compass as we would see it looking aft on a ship steaming due south4 ,it will be seen from this diagram that the phantom ring1 shown in black section1 is extended at the top to form a horizontal flange4 ,on this flange the angular divisions of the compass card are engraved4 ,the phantom ring is also formed with a central hollow stalk whereby it is hung on ball bearings within the central boss of the spider4 ,a torsionless wire composed of several strands of steel is secured at one end to a cap fixed over the top of the stalk1 and at the lower end is secured to a ball-borne pin forming part of the vertical ring4 ,diametrically below1 the vertical ring and the phantom are united by a pivot pin4 ,the gyro case1 as we have explained before1 is carried on a horizontal axis within the vertical ring1 while the bail is swung on the phantom4 ,the phantom and all within it may be regarded as the sensitive element1 the constant alignment of which in the north and south direction gives the device its utility as a compass4 ,the spider carrying the lubber ring and all outside it move round the wire-and-pivot-pin vertical axis with the ship when the course is altered4 ,when such a movement occurs the absence of any direct mechanical connection of a rigid nature between the phantom ring on the one hand and the rest of the sensitive element on the other would tend to result in the phantom ring following the movement of the spider1 binnacle1 and ship rather than in its remaining stationary with the wheel1 casing1 and vertical ring4 ,the suspension wire would thus be twisted4 ,as we have previously explained1 however1 the phantom ring is formed just below the card with a circular rack with which there meshes a pinion driven through reduction gearing by a small electric motor--the azimuth motor--attached to the spider4 ,the starting and stopping of this motor and the direction of its rotation are controlled automatically by means of gold-rimmed wheels on two trolley heads fixed to the vertical ring and certain gold-faced contactors on the phantom4 ,thus when the ship's course is altered the azimuth motor is started up in the direction required to take the twist out of the suspension wire and bring the phantom into alignment with the rest of the sensitive element4 ,when this alignment is reached the motor is automatically cut out4 ,in actual use the phantom overshoots the exact position of alignment by a very small amount1 bringing the reverse contactors into action and starting the motor in the reverse direction4 ,the phantom1 in fact1 oscillates about the position of alignment with the sensitive element through1 roughly1 a quarter of a degree4 ,this small vibration is transmitted to the repeater compasses elsewhere in the ship1 and is of practical value in that its visibility is some assurance that the master compass is working correctly4 ,it will be seen1 then1 that the absence of friction about the vertical axis of the ,sperry compass is secured by suspending the sensitive element within a member--the phantom ring--the frictional drag on which is eliminated by driving it by power in such a way that it follows all the movements of the sensitive element relatively to the supporting spider in an automatic and substantially instantaneous and dead-beat manner4 ,the repeater compasses are operated electrically from the master compass through a transmitter and a pinion meshing with the circular rack on the phantom4 ,the pinion is journalled on a pin depending from the lubber ring1 and therefore transmits to the repeaters not only the relative movement between the phantom and the binnacle1 but1 as we have before explained1 any movement of the lubber ring relatively to the binnacle which may be made to correct the reading of the master compass for the latitude and north steaming errors4 ,in this way the repeater compasses always indicate true north both for course-setting purposes and for the purpose of taking bearings on passing objects4 ,it is regarded as outside the scope of this discussion to describe the mechanical features of the repeater compasses4 ,it may1 however1 be remarked that the fact that they may be fitted in any number and in any position1 and that automatic course recorders and similar devices may also readily be linked up with the gyro-compass1 forms a very strong reason in itself for recommending the adoption of the gyroscopic rather than the magnetic compass on board ship4 ,,chapter ,,xvi ,,the ,,brown ,,compass ,the ,brown compass is the invention of ,mr4 ,s4 ,g4 ,brown1 ,f4,r4,s41 of ,north ,acton1 with ,professor ,john ,perry1 ,f4,r4,s41 in association as technical adviser and co-patentee4 ,it has been evolved after five years of laborious experimental work1 and is claimed to be the only ,british gyroscopic compass so far constructed and applied on board ship4 ,illustration3 ,,fig4 #dg4 ,the ,brown ,compass ,removed from ,binnacle4 ,illustration3 ,,fig4 #dh4 ,the ,brown ,compass ,removed from ,binnacle4 ,illustration3 ,,fig4 #di4 ,the ,brown ,compass4 ,in ,fig4 #dg we show the ,brown compass as seen looking forward on a vessel steaming due north4 ,the south end of the gyro-axle is pointing towards the reader4 ,in ,fig4 #dh it is shown as it would be seen when looking forward on a vessel steaming due west4 ,the north end of the axle is pointing to the right in this view4 ,in the line engraving 7,fig4 #di7 we are supposed to be looking forward on a vessel steaming due south4 ,the north end of the axle is therefore pointing toward us4 ,in the half-tone engravings the compass is shown removed from its external gimbal rings4 ,the axis ,a 7,fig4 #dg71 it will be understood1 is the athwartship axis of the external mounting4 ,it is mounted on ball bearings within a couple of small brackets which by means of four screws are attached to the inner gimbal ring1 as shown in ,fig4 #di4 ,this inner ring1 in turn1 is carried on a longitudinal axis inside an outer gimbal ring1 which1 finally1 is hung on springs from the binnacle4 ,the frame ,b1 within which the sensitive element proper is mounted1 is very much as we have represented it to be in our previous diagrams1 except that it swings pendulum-wise on the axis ,a--or the other axis of the external mounting--not because of the attachment to it at its lowest point of an actual bob weight1 but because its centre of gravity is designedly below the axis of suspension1 and because at its lowest point it carries a casing containing an electric motor and other adjuncts possessing a degree of weight4 ,as shown best in ,fig4 #di1 the frame ,b supports a vertical ring ,c carrying the compass card1 and within which the casing ,d containing the gyro-wheel is mounted on a horizontal1 east and west1 knife-edge axis4 ,the easterly end of this knife-edged axis--towards the left in the engraving--has associated with it the nozzle and divided air box1 whereby1 as we have explained1 the pressure of the air blast generated inside the casing by the spinning wheel is transmitted by open and crossed pipes respectively to the oil-damping bottles ,e and the oil control bottles ,f4 ,what we regard as the most interesting non-gyroscopic feature of the ,brown compass is the manner in which the vertical ring ,c is supported within the frame ,b4 ,the ring at its highest point is formed with a vertical trunnion1 which at its top is guided within a ball bearing fixed in the central boss of the frame ,b4 ,three slip rings lower down on the trunnion acting in conjunction with three mercury contact rings attached to the frame serve to conduct three-phase current to the sensitive element4 ,at the lower side the vertical ring is also provided with a trunnion1 which is of the nature of a foot-step bearing4 ,in reality1 however1 the trunnion is never allowed to touch the bottom of its bearing1 for it--and with it the vertical ring1 the compass card1 and the whole of the sensitive element--is forced1 by means of a supply of oil delivered below it1 to vibrate up and down through about an eighth of an inch at the rate of some #ahj times a minute4 ,before the trunnion can fall to the foot of its bearing it gets a fresh kick up by the oil1 so that no actual metal-to-metal contact is established4 ,the oil supply is drawn from and returned to the reservoir ,g by a pump driven from an electric motor ,h4 ,the slight rapid vibratory motion thus communicated to the sensitive element has no appreciable effect upon the ease or accuracy with which the compass card may be read4 ,on the other hand1 it is sufficient to relieve the frictional resistance to turning at the upper and lower trunnions of the vertical ring in accordance with the well-known fact that if we overcome friction in one direction we overcome it also in the direction at right angles to the first4 ,a second mechanical feature of the ,brown compass of outstanding interest is the method adopted for transmitting the indications of the master compass card to the repeater compasses4 ,the problem to be solved in this matter is to establish connection with the master compass card in a manner which shall impose no frictional or other drag on that card4 ,to this end in the ,brown compass the air blast generated by the spinning wheel is called upon to fulfil a function additional to those to which we have already referred4 ,the air blast is delivered not only from the eastern1 but also from the western end of the horizontal axis of the casing4 ,it is thus directed through a nozzle ,j against the face of a contact maker ,k containing a pair of disc-like plungers working in balanced connection within two cylinders4 ,opposite each plunger is a slot1 and when no movement has to be transmitted to the repeater compasses the air blast is directed against the blank wall between these two slots or enters each slot in equal proportion4 ,thus the plungers being balanced1 neither makes contact4 ,if1 however1 the ship's course is altered1 the contact maker ,k1 being attached to a ring ,l beneath the lubber ring1 moves with the ship1 while the nozzle ,j remains stationary with the sensitive element4 ,thus the air blast exerts unequal pressures on the plungers1 and one of them is forced back to make contact4 ,as a result1 a step-by-step electric motor ,m carried on the frame ,b and in mesh through reduction gearing with a rack on the ring ,l is started up and turns the ring in the direction required to make the two slots again divide the air blast equally4 ,when this neutral position is reached the motor is cut out4 ,as in the ,sperry compass1 the follow-up motion is open to a little hunting movement as a result of the momentum acquired by the parts4 ,ignoring this hunting movement1 the ring ,l and the contact maker ,k are thus driven to remain stationary in fixed relationship with the sensitive element no matter to what extent the frame ,b may move with the ship when the course is altered4 ,the relative movement between the ring ,l and the frame ,b is communicated to the repeater compasses in an electrical manner4 ,the details of the electrical transmission are outside the ambit of this discussion1 but we may say that the movements communicated to the repeater compasses are derived from a distributor on the compass switchboard1 this distributor being arranged to operate in exact synchronism with the follow-up ring ,l4 ,it is claimed for the ,brown system of transmission between the master and the repeater compasses that even should the whole repeater system break down the correct action of the master compass will not thereby be affected4 ,,chapter ,,xvii ,,the ,,ansch,,tz 7#aiab7 ,,compass ,a plan and sectional elevation of the modern form of ,anschtz compass are given in ,figs4 #ej and #ea4 ,the casings of the three gyros ,k ,l ,m hang by vertical stalks below a triangular spider ,a at the centre of which is affixed a float ,b immersed in a bowl ,c containing mercury4 ,after the manner followed in the #aiaj form1 the float and all attached to it are centralised relatively to the bowl by means of a rod ,d fixed centrally to the cover of the compass4 ,as before1 this rod ,d is composed of a central core and a liner insulated from the core4 ,the ends of the core and liner dip into two concentric mercury cups carried within the float in order that two phases of the three-phase current driving the gyro-motors may be transmitted through the core and liner of the rod ,d4 ,the third phase is transmitted through the mercury and float by earthing the bowl ,c4 ,matters are so arranged that the centre of flotation of the float ,b and its attached parts is above the centre of gravity of the floating parts4 ,these floating parts--the spider ,a1 the float ,b1 the three gyros1 and other items not yet mentioned--constitute the sensitive element1 so that this element1 as in the #aiaj form1 does not need the addition of a separate weight to provide the required pendulum action about a horizontal axis through the centre of flotation4 ,instead of there being but one such horizontal axis--the east and west axis ,e ,f of our models and diagrams--it is clear that the support of the sensitive element by means of a float provides the element with an east and west horizontal axis and an infinite number of other horizontal axes4 ,illustration3 ,,fig4 #ej4 ,plan of ,anschtz 7#aiab7 ,compass4 ,attached to the spider ,a--and therefore forming a portion of the sensitive element--is a sheet metal annular casing of the section shown at ,e ,e or ,f ,f4 ,the gyros are enclosed within this casing4 ,ventilating tubes and baffles ,g are provided at points on the casing between each pair of gyros4 ,the compass card--in the form really of a ring--is attached to the casing at ,h4 ,regarding the casing and the spider ,a as the equivalent of the horizontal ring shown in our diagram 7,fig4 #da71 it will be noticed that the gyro casings are not fixed rigidly to it1 but are really mounted on ball bearings surrounding their stalks1 so that they may rotate about a vertical axis relatively to the rest of the sensitive element4 ,as shown in the plan view1 however1 the casing of the gyro ,k is connected by two springs ,j to the rest of the sensitive element1 so that in whichever way the gyro turns on the ball bearings it applies through one or other of the springs a force in the same direction to the annular casing1 etc4 ,the gyro is therefore substantially1 though not actually1 rigidly connected to the rest of the sensitive element1 the springs being introduced to provide a yielding connection which will prevent the full force of a sudden turn of the ship from being thrown all at once on to the gyro4 ,the gyros ,l ,m are similarly connected to the rest of the sensitive element1 but in their case one pair of springs is made to serve both gyros by employing links and a bell-crank lever4 ,the springs1 it may be remarked1 undoubtedly do play a part in the transmission of the directive force from the gyros to the card and in the avoidance of the quadrantal error4 ,but their presence is not essential to the fundamental principle of action of the compass4 ,the damping system of the #aiab design is of a very simple nature1 and represents a great improvement over the air blast method previously used4 ,although the wheels are not required to act as blowers1 their casings are not exhausted of air4 ,the casings1 in fact1 are perforated with four large holes on each side1 the cooling effect of the circulating air being regarded as of more value in practice than the saving of power which would result if the wheels were run in an exhausted atmosphere4 ,the damping force is supplied by the weight of a body of oil contained within a trough ,n extending right round the foot of the annular casing containing the gyros4 ,this trough1 a circle as seen in plan1 is blocked by eight bulkheads1 one each below the north and south points of the compass card1 and the others equally spaced round the trough4 ,through each bulkhead a short pipe passes1 so that the oil in the trough may flow from one compartment to another4 ,with the exception1 however1 of the north and south bulkhead pipes1 which are quite free in the bore1 the passage of the oil is restricted by means of a wire partially filling the bores of the pipes4 ,by varying the size of wire used1 the restriction to the flow of the oil from one compartment to the others1 and therefore the rate at which the oil will flow when the sensitive element tilts1 can be regulated to give the degree of damping required or to suit any change of viscosity in a fresh supply of oil4 ,the system in principle has much in common with the ,brown method of damping4 ,should the compass card suffer an easterly deflection1 the north point of the card will1 as we know1 tend to rise under the influence of the earth's rotation1 and will continue to rise until the turning moment applied by the deflected pendulum weight precesses the card back to the meridian1 whereafter the north point of the card passing over towards the west will begin to fall towards the horizontal plane1 and then1 descending still farther1 it will once more come back to the meridian4 ,during this compound motion the oil in the trough flows backwards and forwards1 accumulating below the south point of the card when the north point of the card is rising and gathering below the north point when the north point is falling4 ,in other words1 there is an excess weight of oil below the southern point of the card throughout the complete half-swing from east to west1 the maximum excess occurring when the card is crossing the meridian4 ,on the half-swing from west to east the excess weight of oil is below the northern point of the card1 the maximum excess occurring1 as before1 when the card is crossing the meridian4 ,the excess weight of oil at all times thus tends to increase the rise or dip of the north point of the card above or below the horizontal plane1 whereas the pendulum weight at all times tends to diminish such rise or dip4 ,hence the excess weight of oil tries to precess the card in the direction opposed to that in which the pendulum weight is precessing it4 ,the vibration of the card in this compass1 as in the ,brown design1 is therefore damped by the generation of a counter-precessional tendency1 and not1 as in the early ,anschtz and the ,sperry designs1 by precessing the sensitive element in the direction required to reduce the angle by which the pendulum weight is tilted away from the plumb line4 ,from what we have already said regarding the ,brown system of damping1 it will readily be inferred that there is no latitude error in the ,anschtz #aiab compass4 ,the damping force is equivalent to a reduction in the weight of the pendulum 8bob10 and is not applied directly to reduce the tilt of the bob4 ,the tilt of the pendulum weight required in north or south latitudes to provide the appropriate rate of westerly or easterly precession is not opposed by the damping force called into play by such tilt4 ,instead1 the damping force merely makes the bob lighter1 so that the tilt has to be carried farther before the effective weight of the bob can balance the tilting action of the earth's rotation4 ,the balance will be automatically struck when the moment of the effective weight of the bob is just sufficient to generate the required rate of westerly or easterly precession appropriate to the latitude4 ,the arrangement of the three gyros at the corners of an equilateral triangle and the general form given to the annular casing and the rest of the sensitive element results in the distribution of the mass of the sensitive element in a very uniform manner around the vertical axis4 ,there is no excessive concentration of the mass towards the east and west plane1 and as a result it is unnecessary to add compensator weights to this compass in order to avoid the effects of centrifugal force during quadrantal rolling4 ,illustration3 ,,fig4 #ea4 ,sectional ,elevation of ,anschtz 7#aiab7 ,compass4 ,the gyro-wheels are made of a special quality of nickel steel1 and are mounted on axles of the de ,laval type--that is to say1 they are tapered and made of very small diameter 7about #j4ae in4 at the parallel ends7--in order that they may yield a little should the centre of gravity of the wheel not be truly coincident with the centre line of the shaft4 ,the wheels are #e in4 in diameter1 weigh #e lb4 #b oz41 and run at #bj1jjj revolutions per minute4 ,the motors are of the squirrel-cage type with the rotor windings fixed to the wheels inside a recess concentric with the axle4 ,the field coils are fixed relatively to the gyro casing4 ,it is of interest to note that when the gyro-wheel is being run up to its full speed--an operation taking about five minutes to complete--the axle passes through three critical speeds4 ,these speeds are approximately #gjjj1 #aa1jjj1 and #ad1jjj revolutions1 and are believed to be associated1 the first with one end of the axle1 the second with the other end1 and the third with a combined action at both ends4 ,during the period of running up the gyros the starting current is1 of course1 heavier than the current taken to drive the wheels at the top speed1 and as a result a considerable temperature is developed in the wheels and their casings4 ,when1 however1 the gyros have been running for some time at the top speed the temperature drops1 and throughout the compass remains fairly constant at about #aej deg4 ,fahr4 ,the viscosity of the damping oil1 on the constancy of which the constancy of the damping force depends1 is therefore less affected by external atmospheric changes of temperature than might be expected4 ,the oil used is a mineral one4 ,it serves not only to damp the vibrations of the card1 but also to lubricate the gyro-axles4 ,to this end1 as shown in ,fig4 #ea1 pipes are led down from each end of each axle to dip into the oil trough1 the flow of oil being induced by means of wicks inside the pipes4 ,the method adopted for the transmission of the readings from the master compass to the repeaters is of considerable interest4 ,the bowl ,c containing the mercury and the float is surrounded by two semi-cylindrical strips ,p ,q of silver-plated brass4 ,at one of the gaps between these strips the two abutting edges are faced with platinum4 ,the gap between these platinum faces is #j4aa in4 in width1 and into it there is inserted1 as shown in the plan view1 a platinum-iridium ball ,r1 measuring #j4jie in4 in diameter4 ,on the switchboard serving the compass there is a reversible motor1 two of the windings of which are constantly connected to a generator--the same generator as serves the gyro-motors4 ,the contact ball ,r is connected to the third phase of the generator1 while the two strips ,p ,q are connected to the third winding of the reversible motor1 this winding being duplicated in such a way that the motor revolves in one direction or the other1 according as the circuit is completed at the ball ,r through the strip ,p or the strip ,q4 ,a commutator is mounted on the axle of the reversible motor1 and from it current is distributed to the motors operating the repeaters and to the 8follow-up0 motor ,s 7,fig4 #ea74 ,the latter motor is geared to the shaft carrying the bowl ,c1 and when started up by the reversible motor turns the bowl in the direction required to restore the ball ,r to the middle of its slot1 and so break the connection with the strip ,p or ,q4 ,thus when the ship's course is altered the bowl tends to rotate with the ship1 but the ball ,r is mounted on the sensitive element1 and therefore maintains its position4 ,contact is thus established between the ball and one of the strips ,p ,q1 the reversible motor is set rotating in the appropriate direction1 and current is distributed to the 8follow-up0 motor ,s to rotate the bowl relatively to the ship until the ball ,r is once more lying midway in the gap4 ,the tendency of the bowl to rotate with the ship is thus counteracted2 the action of the 8follow-up0 motor practically results in the bowl being held in constant relationship to the sensitive element substantially as though it were part thereof4 ,simultaneously the cards of the repeaters are prevented from rotating with the ship1 so that virtually they1 too1 act as if rigidly connected to the sensitive element1 without1 however1 any frictional drag being thrown from them on to the sensitive element4 ,as illustrating the refined construction of the entire compass1 the design of the ball contact may be noticed4 ,the ball is carried at the end of a tapered spiral spring4 ,it is free to rotate on the spring end1 but is prevented from moving axially thereon4 ,the spring end is provided with a button4 ,the ball is drilled out and beaded over the button4 ,to ensure good electrical contact at all times between the ball and the spring a drop of mercury is carried inside the ball between it and the button4 ,should the ship turn very suddenly the ball may spring out of the gap and be dragged across the face of one or other of the strips ,p ,q4 ,it is for this reason that these strips are silver-plated4 ,the repeater compasses are provided not only with an ordinary card graduated from #j deg4 to #afj deg41 but also with an inner dial which makes one revolution for an alteration of #aj deg4 in the ship's course4 ,this dial is graduated to #a_/aj deg41 and permits very small departures from the set course to be immediately noticed and corrected4 ,an elaboration of the same idea is provided in the multiple repeater of the ,brown compass4 ,in this repeater the inner dial is the ordinary #cfj deg4 card4 ,the outer annular dial makes four revolutions for every complete turn of the ship4 ,with the ship sailing due north the graduations on the outer dial are numbered from #j to #de round the east half of the dial1 and from #cfj to #cae round the west half4 ,the numbers1 however1 are not marked on the dial itself1 but on the edges of discs seen through slots in the dial4 ,as the ship turns from the north towards the east1 the discs on the west side of the dial are successively rotated one stage as the south end of the lubber line passes over them1 so as to exhibit numbers forming a continuation of the numbers on the east side of the dial4 ,the outer magnified dial is thus of itself sufficient for navigational purposes4 ,in the ,anschtz equipment arrangements are made for attaching an azimuth mirror to the repeater dial for the purpose of providing an artificial horizon during the taking of bearings4 ,a separate gyroscopically stabilised artificial horizon device1 such as is sometimes to be found on board ships1 is thus rendered unnecessary4 ,,index ,absence of latitude error in ,anschtz 7#aiab7 compass1 #aeh ,absence of latitude error in ,brown compass1 #ff ,action of excentric pin in ,sperry compass1 #ee et seq4 ,air-blast pressure1 ,brown compass1 #fa ,anschtz 7#aiaj7 compass-- ,air-blast damping system1 #db et seq4 ,damping curve1 #ej ,latitude error1 cause of1 #fe -- -- correction for1 #fg -- -- value of1 #fh ,magnitude of directive force at equator1 #bc ,non-gyroscopic details1 #ach ,period of vibration of axle1 #cc1 #cg ,peripheral speed of wheel1 #dg ,quadrantal error1 #if1 #ajg1 #abj ,weight1 diameter1 and speed of wheel1 #bc ,weight of sensitive element1 #cj ,wheel tested to destruction1 #adj ,anschtz 7#aiab7 compass-- ,absence of latitude error1 #aeh ,directive force1 value of1 #abe ,fourth gyro type1 #abh ,gyro-wheels1 details of1 #aei ,non-gyroscopic details1 #aed ,oil damping system1 #aef ,quadrantal error1 #ajg1 #abj1 #abf ,repeater compasses1 #afb ,temperature rise1 #afj ,weight1 diameter1 and speed of wheels1 #abc ,ballistic deflection and error1 #ha et seq4 ,ballistic deflection1 dead-beat1 #hg ,ballistic deflection1 tests by ,british ,admiralty1 #hi ,ballistic gyro1 ,sperry compass1 #aaa ,brown compass-- ,absence of latitude error1 #ff ,air-blast pressure1 #fa ,compensator weights1 #acd ,damping bottles1 #fa -- system1 #ei et seq4 ,generation of directive force1 #aaf ,magnitude of directive force at equator1 #bc ,non-gyroscopic details1 #adh ,oil control bottles1 #aac ,on due west course1 #aag ,quadrantal error1 #aac1 #abf ,repeater compasses1 #aed1 #afb ,weight1 diameter1 and speed of wheel1 #bc ,weight of sensitive element1 #cj ,centrifugal forces during quadrantal rolling1 #acj et seq4 ,clock1 gyroscopic1 #af ,compensator weights1 #acd ,correction mechanism for latitude and north steaming errors1 #ge ,course correction or cosine ring1 #gg ,damped and free motion of gyro-axle1 #di ,damped and undamped vibrations1 #ce et seq4 ,damping curve1 #ej -- systems1 #db1 #dc1 #eb1 #ei1 #aef -- vibrations of gyro-compass1 #bi ,dead-beat ballistic deflection1 #hg ,details of gyro-wheels1 #bc1 #aei ,directive force1 value of1 at equator1 #bc1 #abe ,directive force1 effective1 #bh ,effect of rolling on due west course1 #ic1 #ih1 #ii ,effect of rolling on due north course1 #id ,effect of rolling on north-west course1 #aja ,elementary gyro-compass1 #ah -- -- at equator1 #bj1 #bf -- -- at #ee deg4 ,n4 lat41 #bd -- -- near ,north ,pole1 #bf ,elementary gyroscope at equator1 #ae ,elementary gyroscopic phenomena1 #d et seq4 ,elimination of the quadrantal error1 #ajg ,excentric pin1 action of1 #ee et seq4 -- -- stabilisation of1 #aaj1 #aab ,external gimbal mounting1 #ac1 #ig ,fourth gyro type of ,anschtz compass1 #abh ,free and damped motion of gyro-axle1 #di ,friction1 solid and fluid1 #ci1 #dj ,generation of directive force1 #ah1 #aaf ,german submarines1 #if1 #abj1 #abh ,gimbal mounting1 external1 #ac1 #ig ,gyro-axle1 free and damped motion of1 #di ,gyroscope at equator1 #ae -- with three degrees of freedom1 #d et seq4 ,gyroscope and rotation of the earth1 #ae ,gyroscopic compass at equator1 #bj1 #bf ,gyroscopic compass at #ee deg4 ,n4 lat41 #bd ,gyroscopic compass near ,north ,pole1 #bf ,gyroscopic phenomena1 elementary1 #d et seq4 ,gyro-wheels1 details of1 #bc1 #aei ,latitude bail corrector1 #fi -- corrector dial1 #gf -- error1 #fe -- -- absence of1 #ff1 #aeh -- -- cause of1 #fe1 #ff -- -- correction of1 #fg1 #ge -- -- value of1 #fh1 #fi1 #ge ,magnetic compass1 #a1 #bc1 #cj ,magnitude of directive force at equator1 #bc1 #abe ,north steaming error1 #gj -- -- -- correction mechanism1 #ge -- -- -- value of1 #gd1 #gi ,oil control bottles1 #aac -- damping system1 #aef ,pendulum1 simple1 #ca1 #ha ,period of rolling of a ship1 #ie -- -- vibration of gyro-axle1 #cc1 #cg1 #hh ,peripheral speed of gyro-wheel1 #dg ,phantom ring1 #ec ,quadrantal error1 #ia1 #if1 #ajg1 #abj ,quadrantal error1 elimination of1 #ajg1 #ajh1 #aac1 #abj1 #abf ,quadrantal error1 value of1 #ajg -- rolling1 centrifugal forces during1 #acj et seq4 ,repeater compasses1 #gi1 #hj1 #adf1 #aeb1 #afb ,rolling of a ship1 period of1 #ie -- on due north course1 #ic1 #ih1 #ii -- -- -- west course1 #id -- -- north-west course1 #aja -- quadrantal1 centrifugal forces during1 #acj et seq4 ,rotation of the earth and the gyroscope1 #ae ,simple pendulum1 #ca1 #ha ,sperry compass-- ,bail1 #ed ,ballistic gyro1 #aaa ,compensator weights1 #acd ,course corrector or cosine ring1 #gg ,damping system1 #eb ,directive force1 magnitude at equator1 #bc ,excentric pin1 action of1 #ee et seq4 ,excentric pin1 stabilisation of1 #aaj1 #aab ,latitude bail corrector1 #fi -- corrector dial1 #gf -- error1 cause of1 #ff -- -- correction of1 #fg1 #ge ,latitude error1 value of1 #fi ,non-gyroscopic details1 #adb ,north steaming error1 correction of1 #ge1 et seq4 ,on north-west course1 #ajh ,phantom ring1 #ec ,quadrantal error1 #ajh1 #abf ,repeater compasses1 #adf ,speed correction dial1 #gg ,vacuum in wheel casing1 #eb ,weight1 diameter1 and speed of wheel1 #bc ,weight of sensitive element1 #cj ,submarines1 ,german1 #if1 #abj1 #abh ,temperature rise in ,anschtz compass1 #afj ,test of ,anschtz wheel to destruction1 #adj ,vacuum in ,sperry casing1 #eb ,vibration period of ,anschtz compass1 #cc1 #cg ,vibration period of gyro-axle1 standard1 #hh ,vibrations1 damped and undamped1 #ce et seq4 ,weight1 diameter1 and speed of gyro-wheel1 #bc1 #abc ,weight of sensitive element1 #cj ,printed in ,great ,britain at ,the ,mayflower ,press1 ,plymouth4 ,william ,brendon ,son1 ,ltd4 ,,footnotes #a ,earlier form as in use in #aiaj4 #b ,approximate4 ,differs in different compasses4 #c ,the value of the latitude error is b tan ,l1 where ,l is the angle of latitude and b a constant dependent upon the design of the compass4 ,in the ,sperry correction mechanism the value of the quantity b is represented by the distance between the centre of the dial ,b and the centre of the pin ,c when the radial slot in the dial is lying at right angles to the bar ,d ,e4 #d ,the value of the north steaming error is numerically equal to 7a ,k cos ,c7_/cos ,l1 where ,k is the ship's speed1 ,l the angle of latitude1 ,c the angle between the course and the north and south direction1 and a a constant involving the speed of rotation of the earth4 ,in the ,sperry correction mechanism the value of the quantity a is represented by the radius at which the pin ,l lies from the centre of the latitude dial4 #e ,the ballistic deflection is dependent upon the rate at which the northerly component of the ship's speed is being changed4 ,the difference in the north steaming errors is dependent upon the initial and final values of the speed towards the north1 and would not appear to be affected by the length of time occupied in changing the speed4 ,it would seem1 therefore1 that the ballistic deflection can only be dead-beat in the chosen latitude if the speed towards the north is reduced or increased at one particular rate4 ,thus if the ship is steaming due north at #bj knots and changes its speed to #aj knots in 7a7 #aj minutes or 7b7 #e minutes1 the initial and final values of the north steaming error will be the same in both cases1 since the initial and final speeds are the same1 but the ballistic deflection will be greater in the second case than in the first1 since the rate at which the speed is changed is greater4 ,if1 then1 the ballistic deflection is dead-beat in the first case it cannot be so in the second4 ,the statement that the ballistic deflection in the chosen latitude is dead-beat thus appears1 it may be suggested1 to require qualification4 ,on the other hand1 the ,british ,admiralty in connection with the use of gyro-compasses on destroyers and similar fast-manuvring vessels wherein the ballistic deflection is a factor of very great importance has carried out lengthy experiments on the matter1 and1 although the results have not been divulged1 it is understood that the ballistic deflection was found to be dead-beat independently of the rate at which the speed was changed4 #f ,the wheels in the two designs being similar in form1 their moments of inertia--the real factors determining the magnitude of the directive forces--are proportional to the fourth powers of their diameters4 ,the fourth power of #e is to the fourth power of #f as #a is to #b approximately4 #g ,the directive force is proportional to the sine of the angle of deflection4 ,the gyros ,k ,l ,m therefore supply at #cj deg4 deflection a total directive force of ,d sin #cj deg4 ,d sin #fj deg4 ,d sin #j deg41 or #a4cff ,d4 ,for a single-gyro compass1 having a wheel of twice the inertia of ,k ,l or ,m separately1 the directive force at #cj deg4 deflection would be #b ,d sin #cj deg4 or ,d4 #h ,more precisely1 the moment of inertia of the body must have a constant value about the line ,j ,k1 ,fig4 #dc1 at all angles of setting4 ,transcriber's ,notes ,simple typographical errors were corrected4 ,when necessary1 illustrations have been moved between paragraphs1 so some of the page references in the ,list of ,illustrations are off by a page or two4 ,links to those illustrations are correct4 ,index not checked for proper alphabetization or correct page references4