,,the ,,molecular ,,tactics ,,of ,a ,,crystal ,,lord ,,kelvin ,london ,,henry ,,frowde ,,oxford ,,university ,,press ,,warehouse ,,amen ,,corner1 ,e4,c4 ,illustration ,new ,york ,,macmillan ,,co41 #ff ,,fifth ,,avenue ,,the ,,molecular ,,tactics ,,of ,a ,,crystal ,,by ,,lord ,,kelvin1 ,p4,r4,s4 ,,professor ,,of ,,natural ,,philosophy ,,in ,,the ,,university ,,of ,,glasgow ,,and ,,fellow ,,of ,,peterhouse1 ,,cambridge ,being the ,second ,,robert ,,boyle ,,lecture1 delivered before the ,oxford ,university ,junior ,scientific ,club on ,tuesday1 ,may #af1 #ahic ,,with ,,twenty ,,illustrations ,oxford ,,at ,,the ,,clarendon ,,press #ahid ,oxford ,,printed ,,at ,,the ,,clarendon ,,press ,,by ,,horace ,,hart1 ,,printer ,,to ,,the ,,university ,,on ,,the ,,molecular ,,tactics ,,of ,a ,,crystal ,by ,,lord ,,kelvin1 ,p4,r4,s4 #a4 ,my subject this evening is not the physical properties of crystals1 not even their dynamics2 it is merely the geometry of the structure--the arrangement of the molecules in the constitution of a crystal4 ,every crystal is a homogeneous assemblage of small bodies or molecules4 ,the converse proposition is scarcely true1 unless in a very extended sense of the term crystal 7 #bj below74 ,i can best explain a homogeneous assemblage of molecules by asking you to think of a homogeneous assemblage of people4 ,to be homogeneous every person of the assemblage must be equal and similar to every other3 they must be seated in rows or standing in rows in a perfectly similar manner4 ,each person1 except those on the borders of the assemblage1 must have a neighbour on one side and an equi-distant neighbour on the other3 a neighbour on the left front and an equi-distant neighbour behind on the right1 a neighbour on the right front and an equi-distant neighbour behind on the left4 ,his two neighbours in front and his two neighbours behind are members of two rows equal and similar to the rows consisting of himself and his right-hand and left-hand neighbours1 and their neighbours' neighbours indefinitely to right and left4 ,in particular cases the nearest of the front and rear neighbours may be right in front and right in rear2 but we must not confine our attention to the rectangularly grouped assemblages thus constituted4 ,now let there be equal and similar assemblages on floors above and below that which we have been considering1 and let there be any indefinitely great number of floors at equal distances from one another above and below4 ,think of any one person on any intermediate floor and of his nearest neighbours on the floors above and below4 ,these three persons must be exactly in one line2 this1 in virtue of the homogeneousness of the assemblages on the three floors1 will secure that every person on the intermediate floor is exactly in line with his nearest neighbours above and below4 ,the same condition of alignment must be fulfilled by every three consecutive floors1 and we thus have a homogeneous assemblage of people in three dimensions of space4 ,in particular cases every person's nearest neighbour in the floor above may be vertically over him1 but we must not confine our attention to assemblages thus rectangularly grouped in vertical lines4 #b4 ,consider now any particular person ,c 7,fig4 #a7 on any intermediate floor1 ,d and ,d his nearest neighbours1 ,e and ,e his next nearest neighbours all on his own floor4 ,his next next nearest neighbours on that floor will be in the positions ,f and ,f in the diagram4 ,thus we see that each person ,c is surrounded by six persons1 ,,dd1 ,,ee and ,,ff1 being his nearest1 his next nearest1 and his next next nearest neighbours on his own floor4 ,excluding for simplicity the special cases of rectangular grouping1 we see that the angles of the six equal and similar triangles ,,cde1 ,,cef1 c41 are all acute3 and because the six triangles are equal and similar we see that the three pairs of mutually remote sides of the hexagon ,,defd,e,f are equal and parallel4 ,illustration3 ,,fig4 #a #c4 ,let now ,a1 ,a1 ,a1 c41 denote places of persons of the homogeneous assemblage on the floor immediately above1 and ,b1 ,b1 ,b1 c4 on the floor immediately below1 the floor of ,c4 ,in the diagram let a1 a1 a be points in which the floor of ,,cde is cut by perpendiculars to it through ,a1 ,a1 ,a of the floor above1 and b1 b1 b by perpendiculars from ,b1 ,b1 ,b of the floor below4 ,of all the perpendiculars from the floors immediately above and below1 just two1 one from each1 cut the area of the parallelogram ,,cdef3 and they cut it in points similarly situated in respect to the oppositely oriented triangles into which it is divided by either of its diagonals4 ,hence if a lies in the triangle ,,cde1 the other five triangles of the hexagon must be cut in the corresponding points1 as shown in the diagram4 ,thus1 if we think only of the floor of ,c and of the floor immediately above it1 we have points ,a1 ,a1 ,a vertically above a1 a1 a4 ,imagine now a triangular pyramid1 or tetrahedron1 standing on the base ,,cde and having ,a for vertex3 we see that each of its sides ,,acd1 ,,ade1 ,,aec1 is an acute angled triangle1 because1 as we have already seen1 ,,cde is an acute angled triangle1 and because the shortest of the three distances1 ,,ca1 ,,da1 ,,ea1 is 7 #b7 greater than ,,ce 7though it may be either greater than or less than ,,de74 ,hence the tetrahedron ,,cdea has all its angles acute2 not only the angles of its triangular faces1 but the six angles between the planes of its four faces4 ,this important theorem regarding homogeneous assemblages was given by ,bravais1 to whom we owe the whole doctrine of homogeneous assemblages in its most perfect simplicity and complete generality4 ,similarly we see that we have equal and similar tetrahedrons on the bases ,d,,cf1 ,e,f,c2 and three other tetrahedrons below the floor of ,c1 having the oppositely oriented triangles ,,cd,e1 c4 for their bases and ,b1 ,b1 ,b for their vertices4 ,these three tetrahedrons are equal and heterochirally#a similar to the first three4 ,the consideration of these acute angled tetrahedrons1 is of fundamental importance in respect to the engineering of an elastic solid1 or crystal1 according to ,boscovich4 ,so also is the consideration of the cluster of thirteen points ,c and the six neighbours ,,defd,e,f in the plane of the diagram1 and the three neighbours ,,aa,a on the floor above1 and ,,bb,b on the floor below4 #d4 ,the case in which each of the four faces of each of the tetrahedrons of #c is an equilateral triangle is particularly interesting4 ,an assemblage fulfilling this condition may conveniently be called an #8equilateral homogeneous assemblage1' or1 for brevity1 an #8equilateral assemblage4' ,in an equilateral assemblage ,c's twelve neighbours are all equi-distant from it4 ,i hold in my hand a cluster of thirteen little black balls1 made up by taking one of them and placing the twelve others in contact with it 7and therefore packed in the closest possible order71 and fixing them all together by fish-glue4 ,you see it looks1 in size1 colour1 and shape1 quite like a mulberry4 ,the accompanying diagram shows a stereoscopic view of a similar cluster of balls painted white for the photograph4 ,illustration3 ,,fig4 #b4 #e4 ,by adding ball after ball to such a cluster of thirteen1 and always taking care to place each additional ball in some position in which it is properly in line with others1 so as to make the whole assemblage homogeneous1 we can exercise ourselves in a very interesting manner in the building up of any possible form of crystal of the class called #8cubic' by some writers and #8octahedral' by others4 ,you see before you several examples4 ,i advise any of you who wish to study crystallography to contract with a wood-turner1 or a maker of beads for furniture tassels or for rosaries1 for a thousand wooden balls of about half an inch diameter each4 ,holes through them will do no harm and may even be useful2 but make sure that the balls are as nearly equal to one another1 and each as nearly spherical1 as possible4 ,illustration3 ,,fig4 #c4 #f4 ,you see here before you a large model which ,i have made to illustrate a homogeneous assemblage of points1 on a plan first given1 ,i believe1 by ,mr4 ,william ,barlow 7,nature1 ,december #bj and #bg1 #ahhc74 ,the roof of the model is a lattice-frame 7,fig4 #c7 consisting of two sets of eight parallel wooden bars crossing one another1 and kept together by pins through the middles of the crossings4 ,as you see1 ,i can alter it to make parallelograms of all degrees of obliquity till the bars touch1 and again you see ,i can make them all squares4 #g4 ,the joint pivots are 7for cheapness of construction7 of copper wire1 each bent to make a hook below the lattice frame4 ,on these sixty-four hooks are hung sixty-four fine cords1 firmly stretched by little lead weights4 ,each of these cords 7,fig4 #d7 bears eight short perforated wooden cylinders1 which may be slipped up and down to any desired position#b4 ,they are at present actually placed at distances consecutively each equal to the distance from joint to joint of the lattice frame4 ,illustration3 ,,fig4 #d4 #h4 ,the roof of the model is hung by four cords1 nearly vertical1 of independently variable lengths1 passing over hooks from fixed points above1 and kept stretched by weights1 each equal to one quarter of the weight of roof and pendants4 ,you see now by altering the angles of the lattice work and placing it horizontal or in any inclined plane1 as ,i am allowed to do readily by the manner in which it is hung1 ,i have three independent variables1 by varying which ,i can show you all varieties of homogeneous assemblages1 in which three of the neighbours of every point are at equal distances from it4 ,you see here1 for example1 we have the equilateral assemblage4 ,i have adjusted the lattice roof to the proper angle1 and its plane to the proper inclination to the vertical1 to make a wholly equilateral assemblage of the little cylinders of wood on the vertical cords1 a case1 as we have seen1 of special importance4 ,if ,i vary also the distances between the little pieces of wood on the cords2 and the distances between the joints of the lattice work 7variations easily understood1 though not conveniently producible in one model without more of mechanical construction than would be worth making71 ,i have three other independent variables4 ,by properly varying these six independent variables1 three angles and three lengths1 we may give any assigned value to each edge of one of the fundamental tetrahedrons of #c4 #i4 ,our assemblage of people would not be homogeneous unless its members were all equal and similar and in precisely similar attitudes1 and were all looking the same way4 ,you understand what a number of people seated or standing on a floor or plain and looking the same way means4 ,but the expression #8looking' is not conveniently applicable to things that have no eyes1 and we want a more comprehensive mode of expression4 ,we have it in the words #8orientation1' #8oriented1' and 7verb7 #8to orient1' suggested by an extension of the idea involved in the word #8orientation1' first used to signify positions relatively to east and west of ancient ,greek and ,egyptian temples and ,christian churches4 ,but for the orientation of a house or temple we have only one angle1 and that angle is called #8azimuth' 7the name given to an angle in a horizontal plane74 ,for orientation in three dimensions of space we must extend our ideas and consider position with reference to east and west and up and down4 ,a man lying on his side with his head to the north and looking east1 would not be similarly oriented to a man standing upright and looking east4 ,to provide for the complete specification of how a body is oriented in space we must have in the body a plane of reference1 and a line of reference in this plane1 belonging to the body and moving with it4 ,we must also have a fixed plane and a fixed line of reference in it1 relatively to which the orientation of the moveable body is to be specified2 as1 for example1 a horizontal plane and the east and west horizontal line in it4 ,the position of a body is completely specified when the angle between the plane of reference belonging to it1 and the fixed plane is given2 and when the angles between the line of intersection of the two planes and the lines of reference in them are also given4 ,thus we see that three angles are necessary and sufficient to specify the orientation of a moveable body1 and we see how the specification is conveniently given in terms of three angles4 #aj4 ,to illustrate this take a book lying on the table before you with its side next the title-page up1 and its back to the north4 ,i now lift the east edge 7the top of the book71 keeping the bottom edge north and south on the table till the book is inclined1 let us say1 #bj to the table4 ,next1 without altering this angle of #bj1 between the side of the book and the table1 ,i turn the book round a vertical axis1 through #de till the bottom edge lies north-east and south-west4 ,lastly1 keeping the book in the plane to which it has been thus brought1 ,i turn it round in this plane through #ce4 ,these three angles of #bj1 #de1 and #ce1 specify1 with reference to the horizontal plane of the table and the east and west line in it1 the orientation of the book in the position to which you have seen me bring it1 and in which ,i hold it before you4 #aa4 ,in ,figs4 #e and #f you see two assemblages1 each of twelve equal and similar molecules in a plane4 ,fig4 #e1 in which the molecules are all same-ways oriented1 is one homogeneous assemblage of twenty-four molecules4 ,fig4 #f1 in which in one set of rows the molecules are alternately oriented two different ways1 may either be regarded as two homogeneous assemblages1 each of twelve single molecules2 or one homogeneous assemblage of twelve pairs of those single molecules4 ,illustration3 ,,fig4 #e4 #ab4 ,i must now call your attention to a purely geometrical question#c of vital interest with respect to homogeneous assemblages in general1 and particularly the homogeneous assemblage of molecules constituting a crystal3--what can we take as #8the' boundary or #8a' boundary enclosing each molecule with whatever portion of space around it we are at liberty to choose for it1 and separating it from neighbours and their portions of space given to them in homogeneous fairness8 ,illustration3 ,,fig4 #f4 #ac4 ,if we had only mathematical points to consider we should be at liberty to choose the simple obvious partitioning by three sets of parallel planes4 ,even this may be done in an infinite number of ways1 thus3--,beginning with any point ,p of the assemblage1 choose any other three points ,a1 ,b1 ,c1 far or near1 provided only that they are not in one plane with ,p1 and that there is no other point of the assemblage in the lines ,,pa1 ,,pb1 ,,pc1 or within the volume of the parallelepiped of which these lines are conterminous edges1 or within the areas of any of the faces of this parallelepiped4 ,there will be points of the assemblage at each of the corners of this parallelepiped and at all the corners of the parallelepipeds equal and similar to it which we find by drawing sets of equi-distant planes parallel to its three pairs of faces4 7,a diagram is unnecessary47 ,every point of the assemblage is thus at the intersection of three planes1 which is also the point of meeting of eight neighbouring parallelepipeds4 ,shift now any one of the points of the assemblage to a position within the volume of any one of the eight parallelepipeds1 and give equal parallel motions to all the other points of the assemblage4 ,thus we have every point in a parallelepipedal cell of its own1 and all the points of the assemblage are similarly placed in their cells1 which are themselves equal and similar4 #ad4 ,but now if1 instead of a single point for each member of the assemblage1 we have a group of points1 or a globe or cube or other geometrical figure1 or an individual of a homogeneous assemblage of equal1 similar1 similarly dressed1 and similarly oriented ladies1 sitting in rows1 or a homogeneous assemblage of trees closely planted in regular geometrical order on a plane with equal and similar distributions of molecules1 and parallel planes above and below1 we may find that the best conditioned plane-faced parallelepipedal partitioning which we can choose would cut off portions properly belonging to one molecule of the assemblage and give them to the cells of neighbours4 ,to find a cell enclosing all that belongs to each individual1 for example1 every part of each lady's dress1 however complexly it may be folded among portions of the equal and similar dresses of neighbours2 or1 every twig1 leaf1 and rootlet of each one of the homogeneous assemblage of trees2 we must alter the boundary by give-and-take across the plane faces of the primitive parallelepipedal cells1 so that each cell shall enclose all that belongs to one molecule1 and therefore 7because of the homogeneousness of the partitioning7 nothing belonging to any other molecule4 ,the geometrical problem thus presented1 wonderfully complex as it may be in cases such as some of those which ,i have suggested1 is easily performed for any possible case if we begin with any particular parallelepipedal partitioning determined for corresponding points of the assemblage as explained in #ac1 for any homogeneous assemblage of single points4 ,we may prescribe to ourselves that the corners are to remain unchanged1 but if so they must to begin with either in interfaces of contact between the individual molecules1 or in vacant space among the molecules4 ,if this condition is fulfilled for one corner it is fulfilled for all1 as the corners are essentially corresponding points relatively to the assemblage4 #ae4 ,begin now with any one of the twelve straight lines between corners which constitute the twelve edges of the parallelepiped1 and alter it arbitrarily to any curved or crooked line between the same pair of corners1 subject only to the conditions 7#a7 that it does not penetrate the substance of any member of the assemblage1 and 7#b7 that it is not cut by equal and similar parallel curves#d between other pairs of corners4 ,illustration3 ,,fig4 #g4 ,considering now the three fours of parallel edges of the parallelepiped1 let the straight lines of one set of four be altered to equal and similar parallel curves in the manner which ,i have described2 and proceed by the same rule for the other two sets of four edges4 ,we thus have three fours of parallel curved edges instead of the three fours of parallel straight edges of our primitive parallelepiped with corners 7each a point of intersection of three edges7 unchanged4 ,take now the quadrilateral of four curves substituted for the four straight edges of one face of the parallelepiped4 ,we may call this quadrilateral a curvilineal parallelogram1 because it is a circuit composed of two pairs of equal parallel curves4 ,draw now a curved surface 7an infinitely thin sheet of perfectly extensible india-rubber if you please to think of it so7 bordered by the four edges of our curvilineal parallelogram1 and so shaped as not to cut any of the substance of any molecule of the assemblage4 ,do the same thing with an exactly similar and parallel sheet relatively to the opposite face of the parallelepiped2 and again the same for each of the two other pairs of parallel faces4 ,we thus have a curved-faced parallelepiped enclosing the whole of one molecule and no part of any other2 and by similar procedure we find a similar boundary for every other molecule of the assemblage4 ,each wall of each of these cells is common to two neighbouring molecules1 and there is no vacant space anywhere between them or at corners4 ,fig4 #g illustrates this kind of partitioning by showing a plane section parallel to one pair of plane faces of the primitive parallelepiped1 for an ideal case4 ,the plane diagram is in fact a realization of the two-dimensional problem of partitioning the pine pattern of a ,persian carpet by parallelograms about as nearly rectilinear as we can make them4 ,in the diagram faint straight lines are drawn to show the primitive parallelogrammatic partitioning4 ,it will be seen that of all the crossings 7marked with dots in the diagram7 every one is similarly situated to every other in respect to the homogeneously repeated pattern figures3 ,a1 ,b1 ,c1 ,d are four of them at the corners of one cell4 #af4 ,confining our attention for a short time to the homogeneous division of a plane1 remark that the division into parallelograms by two sets of crossing parallels is singular in this respect--each cell is contiguous with three neighbours at every corner4 ,any shifting1 large or small1 of the parallelograms by relative sliding in one direction or another violates this condition1 brings us to a configuration like that of the faces of regularly hewn stones in ordinary bonded masonry1 and gives a partitioning which fulfils the condition that at each corner each cell has only two neighbours4 ,each cell is now virtually a hexagon1 as will be seen by the letters ,a1 ,b1 ,c1 ,d1 ,e1 ,f in the diagram ,fig4 #h4 ,a and ,d are to be reckoned as corners1 each with an interior angle of #ahj4 ,in this diagram the continuous heavy lines and the continuous faint lines crossing them show a primitive parallelogrammatic partition by two sets of continuous parallel intersecting lines4 ,the interrupted crossing lines 7heavy7 show1 for the same homogeneous distribution of single points or molecules1 the virtually hexagonal partitioning which we get by shifting the boundary from each portion of one of the light lines to the heavy line next it between the same continuous parallels4 ,illustration3 ,,fig4 #h4 ,fig4 #h bis represents a further modification of the boundary by which the #ahj angles ,a1 ,d1 become angles of less than #ahj4 ,the continuous parallel lines 7light7 and the short light portions of the crossing lines show the configuration according to ,fig4 #h1 from which this diagram is derived4 #ag4 ,in these diagrams 7,figs4 #h and #h bis7 the object enclosed is small enough to be enclosable by a primitive parallelogrammatic partitioning of two sets of continuous crossing parallel straight lines1 and by the partitioning of #8bonded' parallelograms both represented in ,fig4 #h1 and by the derived hexagonal partitioning represented in ,fig4 #h bis1 with faint lines showing the primitive and the secondary parallelograms4 ,in ,fig4 #g the objects enclosed were too large to be enclosable by any rectilinear parallelogrammatic or hexagonal partitioning4 ,the two sets of parallel faint lines in ,fig4 #g show a primitive parallelogrammatic partitioning and the corresponding pairs of parallel curves intersecting at the corners of these parallelograms1 of which ,a1,b1,c1,d is a specimen1 show a corresponding partitioning by curvilineal parallelograms4 ,fig4 #i shows for the same homogeneous distribution of objects a better conditioned partitioning1 by hexagons in each of which one pair of parallel edges is curved4 ,the sets of intersecting parallel straight lines in ,fig4 #i show the same primitive parallelogrammatic partitioning as in ,fig4 #g1 and the same slightly shifted to suit points chosen for well-conditionedness of hexagonal partitioning4 ,illustration3 ,,fig4 #h bis4 ,illustration3 ,,fig4 #i4 #ah4 ,for the division of continuous three-dimensional space#e into equal1 similar1 and similarly oriented cells1 quite a corresponding transformation from partitioning by three sets of continuous mutually intersecting parallel planes to any possible mode of homogeneous partitioning1 may be investigated by working out the three-dimensional analogue of #af-ag4 ,thus we find that the most general possible homogeneous partitioning of space with plane interfaces between the cells gives us fourteen walls to each cell1 of which six are three pairs of equal and parallel parallelograms1 and the other eight are four pairs of equal and parallel hexagons1 each hexagon being bounded by three pairs of equal and parallel straight lines4 ,this figure1 being bounded by fourteen plane faces1 is called a tetrakaidekahedron4 ,it has thirty-six edges of intersection between faces2 and twenty-four corners1 in each of which three faces intersect4 ,a particular case of it1 which ,i call an orthic tetrakaidekahedron1 being that in which the six parallelograms are equal squares1 the eight hexagonal faces are equal equilateral and equiangular hexagons1 and the lines joining corresponding points in the seven pairs of parallel faces are perpendicular to the planes of the faces1 is represented by a stereoscopic picture in ,fig4 #aj4 ,the thirty-six edges and the twenty-four corners1 which are easily counted in this diagram1 occur in the same relative order in the most general possible partitioning1 whether by plane-faced tetrakaidekahedrons or by the generalized tetrakaidekahedron described in #ai4 #ai4 ,the most general homogeneous division of space is not limited to plane-faced cells2 but it still consists essentially of tetrakaidekahedronal cells1 each bounded by three pairs of equal and parallel quadrilateral faces1 and four pairs of equal and parallel hexagonal faces1 neither the quadrilaterals nor the hexagons being necessarily plane4 ,each of the thirty-six edges may be straight or crooked or curved2 the pairs of opposite edges1 whether of the quadrilaterals or hexagons1 need not be equal and parallel2 neither the four corners of each quadrilateral nor the six corners of each hexagon need be in one plane4 ,but every pair of corresponding edges of every pair of parallel corresponding faces1 whether quadrilateral or hexagonal1 must be equal and parallel4 ,i have described an interesting case of partitioning by tetrakaidekahedrons of curved faces with curved edges in a paper#f published about seven years ago4 ,in this case each of the quadrilateral faces is plane4 ,each hexagonal face is a slightly curved surface having three rectilineal diagonals through its centre in one plane4 ,illustration3 ,,fig4 #aj4 ,the six sectors of the face between these diagonals lie alternately on opposite sides of their plane1 and are bordered by six arcs of plane curves lying on three pairs of parallel planes4 ,this tetrakaidekahedronal partitioning fulfils the condition that the angles between three planes meeting in an edge are everywhere each #abj2 a condition that cannot be fulfilled in any plane-faced tetrakaidekahedron4 ,each hexagonal wall is an anticlastic surface of equal opposite curvatures at every point1 being the surfaces of minimum area bordered by six curved edges4 ,it is shown easily and beautifully1 and with a fair approach to accuracy1 by choosing six little circular arcs of wire1 and soldering them together by their ends in proper planes for the six edges of the hexagon2 and dipping it in soap solution and taking it out4 #bj4 ,returning now to the tactics of a homogeneous assemblage1 remark that the qualities of the assemblage as a whole depend both upon the character and orientation of each molecule1 and on the character of the homogeneous assemblage formed by corresponding points of the molecules4 ,after learning the simple mathematics of crystallography1 with its indicial system#g for defining the faces and edges of a crystal according to the ,bravais rows and nets and tetrahedrons of molecules in which we think only of a homogeneous assemblage of points1 we are apt to forget that the true crystalline molecule1 whatever its nature may be1 has sides1 and that generally two opposite sides of each molecule may be expected to be very different in quality1 and we are almost surprised when mineralogists tell us that two parallel faces on two sides of a crystal have very different qualities in many natural crystals4 ,we might almost as well be surprised to find that an army in battle array1 which is a kind of large-grained crystal1 presents very different appearance to any one looking at it from outside1 according as every man in the ranks with his rifle and bayonet faces to the front or to the rear or to one flank or to the other4 #ba4 ,consider1 for example1 the ideal case of a crystal consisting of hard equal and similar tetrahedronal solids all same-ways oriented4 ,a thin plate of crystal cut parallel to any one set of the faces of the constituent tetrahedrons would have very different properties on its two sides2 as the constituent molecules would all present points outwards on one side and flat surfaces on the other4 ,we might expect that the two sides of such a plate of crystal would become oppositely electrified when rubbed by one and the same rubber2 and1 remembering that a piece of glass with part of its surface finely ground but not polished and other parts polished becomes1 when rubbed with white silk1 positively electrified over the polished parts and negatively electrified over the non-polished parts1 we might almost expect that the side of our supposed crystalline plate towards which flat faces of the constituent molecules are turned would become positively electrified1 and the opposite side1 showing free molecular corners1 would become negatively electrified1 when both are rubbed by a rubber of intermediate electric quality4 ,we might also from elementary knowledge of the fact of piezo-electricity1 that is to say1 the development of opposite electricities on the two sides of a crystal by pressure1 expect that our supposed crystalline plate1 if pressed perpendicularly on its two sides1 would become positively electrified on one of them and negatively on the other4 #bb4 ,intimately connected with the subject of enclosing cells for molecules of given shape1 assembled homogeneously1 is the homogeneous packing together of equal and similar molecules of any given shape4 ,in every possible case of any infinitely great number of similar bodies the solution is a homogeneous assemblage4 ,but it may be a homogeneous assemblage of single solids all oriented the same way1 or it may be a homogeneous assemblage of clusters of two or more of them placed together in different orientations4 ,for example1 let the given bodies be halves 7oblique or not oblique7 of any parallelepiped on the two sides of a dividing plane through a pair of parallel edges4 ,the two halves are homochirally#h similar2 and1 being equal1 we may make a homogeneous assemblage of them by orienting them all the same way and placing them properly in rows4 ,but the closest packing of this assemblage would necessarily leave vacant spaces between the bodies3 and we get in reality the closest possible packing of the given bodies by taking them in pairs oppositely oriented and placed together to form parallelepipeds4 ,these clusters may be packed together so as to leave no unoccupied space4 ,whatever the number of pieces in a cluster in the closest possible packing of solids may be for any particular shape1 we may consider each cluster as itself a given single body1 and thus reduce the problem to the packing closely together of assemblages of individuals all sameways oriented2 and to this problem therefore it is convenient that we should now confine our attention4 #bc4 ,to avoid complexities such as those which we find in the familiar problem of homogeneous packing of forks or spoons or tea-cups or bowls1 of any ordinary shape1 we shall suppose the given body to be of such shape that no two of them similarly oriented can touch one another in more than one point4 ,wholly convex bodies essentially fulfil this condition2 but it may also be fulfilled by bodies not wholly convex1 as is illustrated in ,fig4 #aa4 ,illustration3 ,fig4 #aa4 #bd4 ,to find close and closest packing of any number of our solids ,s#a1 ,s#b1 ,s#c 444 of shape fulfilling the condition of #bc proceed thus3-- 7#a7 ,bring ,s#b to touch ,s#a at any chosen point p of its surface 7,fig4 #ab74 7#b7 ,bring ,s#c to touch ,s#a and ,s#b1 at r and q respectively4 7#c7 ,bring ,s#d 7not shown in the diagram7 to touch ,s#a1 ,s#b1 and ,s#c4 7#d7 ,place1 any number of the bodies together in three rows continuing the lines of ,s#a,s#b1 ,s#a,s#c1 ,s#a,s#d1 and in three sets of equi-distant rows parallel to these4 ,this makes a homogeneous assemblage4 ,in the assemblage so formed the molecules are necessarily found to be in three sets of rows parallel respectively to the three pairs ,s#b,s#c1 ,s#c,s#d1 ,s#d,s#b4 ,the whole space occupied by an assemblage of n of our solids thus arranged has clearly #fn times the volume of a tetrahedron of corresponding points of ,s#a1 ,s#b1 ,s#c1 ,s#d4 ,hence the closest of the close packings obtained by the operations 7#a7 444 7#d7 is found if we perform the operations 7#a71 7#b71 and 7#c7 as to make the volume of this tetrahedron least possible4 ,illustration3 ,,fig4 #ab #be4 ,it is to be remarked that operations 7#a7 and 7#b7 leave for 7#c7 no liberty of choice for the place of ,s#d1 except between two determinate positions on opposite sides of the group ,s#a1 ,s#b1 ,s#c4 ,the volume of the tetrahedron will generally be different for these two positions of ,s#d1 and1 even if the volume chance to be equal in any case1 we have differently shaped assemblages according as we choose one or other of the two places for ,s#d4 ,this will be understood by looking at ,fig4 #ab1 showing ,s#a and neighbours on each side of it in the rows of ,s#a,s#b1 ,s#a,s#c1 and in a row parallel to that of ,s#b,s#c4 ,the plane of the diagram is parallel to the planes of corresponding points of these seven bodies1 and the diagram is a projection of these bodies by lines parallel to the intersections of the tangent planes through p and r4 ,if the three tangent planes through p1 q1 and r1 intersected in parallel lines1 q would be seen like p and r as a point of contact between the outlines of two of the bodies2 but this is only a particular case1 and in general q must1 as indicated in the diagram1 be concealed by one or other of the two bodies of which it is the point of contact4 ,now imagining1 to fix our ideas and facilitate brevity of expression1 that the planes of corresponding points of the seven bodies are horizontal1 we see clearly that ,s#d may be brought into proper position to touch ,s#a1 ,s#b1 and ,s#c either from above or from below2 and that there is one determinate place for it if we bring it into position from above1 and another determinate place for it if we bring it from below4 #bf4 ,if we look from above at the solids of which ,fig4 #ab shows the outline1 we see essentially a hollow leading down to a perforation between ,s#a1 ,s#b1 ,s#c1 and if we look from below we see a hollow leading upwards to the same perforation3 this for brevity we shall call the perforation pqr4 ,the diagram shows around ,s#a six hollows leading down to perforations1 of which two are similar to pqr1 and the other three1 of which pqr indicates one1 are similar one to another but are dissimilar to pqr4 ,if we bring ,s#d from above into position to touch ,s#a1 ,s#b1 and ,s#c1 its place thus found is in the hollow pqr1 and the places of all the solids in the layer above that of the diagram are necessarily in the hollows similar to pqr4 ,in this case the solids in the layer below that of the diagram must lie in the hollows below the perforations dissimilar to pqr1 in order to make a single homogeneous assemblage4 ,in the other case1 ,s#d brought up from below finds its place on the under side of the hollow pqr1 and all solids of the lower layer find similar places3 while solids in the layer above that of the diagram find their places in the hollows similar to pqr4 ,in the first case there are no bodies of the upper layer in the hollows above the perforations similar to pqr1 and no bodies of the lower layer in the hollows below the perforations similar to pqr4 ,in the second case there are no bodies of the upper layer in the hollows above the perforations similar to pqr1 and none of the under layer in the hollows below the perforations similar to pqr4 #bg4 ,going back now to operation 7#a7 of #bc1 remark that when the point of contact p is arbitrarily chosen on one of the two bodies ,s#a1 the point of contact on the other will be the point on it corresponding to the point or one of the points of ,s#a1 where its tangent plane is parallel to the tangent plane at p4 ,if ,s#a is wholly convex it has only two points at which the tangent planes are parallel to a given plane1 and therefore the operation 7#a7 is determinate and unambiguous4 ,but if there is any concavity there will be four or some greater even number of tangent planes parallel to any one of some planes1 while there will be other planes to each of which only one pair of tangent planes is parallel4 ,hence1 operation 7#a71 though still determinate1 will have a multiplicity of solutions1 or only a single solution1 according to the choice made of the position of p4 ,henceforth however1 to avoid needless complications of ideas1 we shall suppose our solids to be wholly convex2 and of some such unsymmetrical shape as those indicated in ,fig4 #ab of #be1 and shown by stereoscopic photograph in ,fig4 #ac of #cf4 ,with or without this convenient limitation1 operation 7#a7 has two freedoms1 as p may be chosen freely on the surface of ,s#a2 and operation 7#b7 has clearly just one freedom after operation 7#a7 has been performed4 ,thus1 for a solid of any given shape1 we have three disposables1 or1 as commonly called in mathematics1 three #8independent variables1' all free for making a homogeneous assemblage according to the rule of #bb4 #bh4 ,in the homogeneous assemblage defined in #bd1 each solid1 ,s#a1 is touched at twelve points1 being the three points of contact with ,s#b1 ,s#c1 ,s#d1 and the three #c's of points on ,s#a corresponding to the points on ,s#b1 ,s#c1 ,s#d1 at which these bodies are touched by the others of the quartet4 ,this statement is somewhat difficult to follow1 and we see more clearly the twelve points of contact by not confining our attention to the quartet ,s#a1 ,s#b1 ,s#c1 ,s#d 7convenient as this is for some purposes71 but completing the assemblage and considering six neighbours around ,s#a in one plane layer of the solids as shown in ,fig4 #ab1 with their six points prqprq of contact with ,s#a2 and the three neighbours of the two adjacent parallel layers which touch it above and below4 ,this cluster of thirteen1 ,s#a and twelve neighbours1 is shown for the case of spherical bodies in the stereoscopic photograph of #d above4 ,we might of course1 if we pleased1 have begun with the plane layer of which ,s#a1 ,s#b1 ,s#d are members1 or with that of which ,s#a1 ,s#c1 ,s#d are members1 or with the plane layer parallel to the fourth side ,s#b ,s#c ,s#d of the tetrahedron3 and thus we have four different ways of grouping the twelve points of contact on ,s#a into one set of six and two sets of three4 #bi4 ,in this assemblage we have what ,i call #8close order' or #8close packing4' ,for closest of close packings the volume of the tetrahedron 7 #bd7 of corresponding points of ,s#a1 ,s#b1 ,s#c1 and ,s#d must be a minimum1 and the least of minimums if1 as generally will be the case1 there are two more different configurations for each of which the volume is a minimum4 ,there will in general also be configurations of minimax volume and of maximum volume1 subject to the condition that each body is touched by twelve similarly oriented neighbours4 #cj4 ,pause for a moment to consider the interesting kinematical and dynamical problems presented by a close homogeneous assemblage of smooth solid bodies of given convex shape1 whether perfectly frictionless or exerting resistance against mutual sliding according to the ordinarily stated law of friction between dry hard solid bodies4 ,first imagine that they are all similarly oriented and each in contact with twelve neighbours1 except outlying individuals 7which there must be at the boundary if the assemblage is finite1 and each of which is touched by some number of neighbours less than twelve74 ,the coherent assemblage thus defined constitutes a kinematic frame or skeleton for an elastic solid of very peculiar properties4 ,instead of the six freedoms1 or disposables1 of strain presented by a natural solid it has only three4 ,change of shape of the whole can only take place in virtue of rotation of the constituent parts relatively to any one chosen row of them1 and the plane through it and another chosen row4 #ca4 ,suppose first the solids to be not only perfectly smooth but perfectly frictionless4 ,let the assemblage be subjected to equal positive or negative pressure inwards all around its boundary4 ,every position of minimum1 minimax1 or maximum volume will be a position of equilibrium4 ,if the pressure is positive the equilibrium will be stable if1 and unstable unless1 the volume is a minimum4 ,if the pressure is negative the equilibrium will be stable if1 and unstable unless1 the volume is a maximum4 ,configurations of minimax volume will be essentially unstable4 #cb4 ,consider now the assemblage of #ca in a position of stable equilibrium under the influence of a given constant uniform pressure inwards all round its boundary4 ,it will have rigidity in simple proportion to the amount of this pressure4 ,if now by the superposition of non-uniform pressure at the boundary1 for example equal and opposite pressures on two sides of the assemblage1 a finite change of shape is produced3 the whole assemblage essentially swells in bulk4 ,this is the #8dilatancy' which ,osborne ,reynolds has described#i in an exceedingly interesting manner with reference to a sack of wheat or sand1 or an india-rubber bag tightly filled with sand or even small shot4 ,consider1 for example1 a sack of wheat filled quite full and standing up open4 ,it is limp and flexible4 ,now shake it down well1 fill it quite full1 shake again1 so as to get as much into it as possible1 and tie the mouth very tightly close4 ,the sack becomes almost as stiff as a log of wood of the same shape4 ,open the mouth partially1 and it becomes again limp1 especially in the upper parts of the bag4 ,in ,reynolds' observations on india-rubber bags of small shot his #8dilatancy' depends1 essentially and wholly1 on breaches of some of the contacts which exist between the molecules in their configuration of minimum volume3 and it is possible that in all his cases the dilatations which he observed are chiefly1 if not wholly1 due to such breaches of contact4 ,but it is possible1 it almost seems probable1 that in bags or boxes of sand or powder1 of some kinds of smooth rounded bodies of any shape1 not spherical or ellipsoidal1 subjected persistently to unequal pressures in different directions1 and well shaken1 stable positions of equilibrium are found with almost all the particles each touched by twelve others4 ,here is a curious subject of ,natural ,history through all ages till #ahhe1 when ,reynolds brought it into the province of ,natural ,philosophy by the following highly interesting statement3--#8,a well-marked phenomenon receives its explanation at once from the existence of dilatancy in sand4 ,when the falling tide leaves the sand firm1 as the foot falls on it the sand whitens and appears momentarily to dry round the foot4 ,when this happens the sand is full of water1 the surface of which is kept up to that of the sand by capillary attractions2 the pressure of the foot causing dilatation of the sand more water is required1 which has to be obtained either by depressing the level of the surface against the capillary attractions1 or by drawing water through the interstices of the surrounding sand4 ,this latter requires time to accomplish1 so that for the moment the capillary forces are overcome2 the surface of the water is lowered below that of the sand1 leaving the latter white or drier until a sufficient supply has been obtained from below1 when the surface rises and wets the sand again4 ,on raising the foot it is generally seen that the sand under the foot and around becomes momentarily wet2 this is because1 on the distorting forces being removed1 the sand again contracts1 and the excess of water finds momentary relief at the surface4' ,this proves that the sand under the foot1 as well as the surface around it1 must be dry for a short time after the foot is pressed upon it1 though we cannot see it whitened1 as the foot is not transparent4 ,that it is so has been verified by ,mr4 ,alex4 ,galt1 ,experimental ,instructor in the ,physical ,laboratory of ,glasgow ,university1 by laying a small square of plate-glass on wet sand on the sea-shore of ,helensburgh1 and suddenly pressing on it by a stout stick with nearly all his weight4 ,he found the sand1 both under the glass and around it in contact with the air1 all became white at the same moment4 ,of all the two hundred thousand million men1 women1 and children who1 from the beginning of the world1 have ever walked on wet sand1 how many1 prior to the ,british ,association ,meeting at ,aberdeen in #ahhe1 if asked1 #8,is the sand compressed under your foot8' would have answered otherwise than #8,yes6'8 7,contrast with this the case of walking over a bed of wet sea-weed67 #cc4 ,in the case of globes packed together in closest order 7and therefore also in the case of ellipsoids1 if all similarly oriented71 our condition of coherent contact between each molecule and twelve neighbours implies absolute rigidity of form and constancy of bulk4 ,hence our convex solid must be neither ellipsoidal nor spherical in order that there may be the changes of form and changes of bulk which we have been considering as dependent on three independent variables specifying the orientation of each solid relatively to rows of the assemblage4 ,an interesting dynamical problem is presented by supposing any mutual forces1 such as might be produced by springs1 to act between the solid molecules1 and investigating configurations of equilibrium on the supposition of frictionless contacts4 ,the solution of it of course is that the potential energy of the springs must be a minimum or a minimax or a maximum for equilibrium1 and a minimum for stable equilibrium4 ,the solution will be a configuration of minimum or minimax1 or maximum1 volume1 only in the case of pressure equal in all directions4 #cd4 ,a purely geometrical question1 of no importance in respect to the molecular tactics of a crystal but of considerable interest in pure mathematics1 is forced on our attention by our having seen 7 #bg7 that a homogeneous assemblage of solids of given shape1 each touched by twelve neighbours1 has three freedoms which may be conveniently taken as the three angles specifying the orientation of each molecule relatively to rows of the assemblage as explained in #cj4 ,consider a solid ,s#a and the twelve neighbours which touch it1 and try if it is possible to cause it to touch more than twelve of the bodies4 ,attach ends of three thick flexible wires to any places on the surface of ,s#a2 carry the wires through interstices of the assemblage1 and attach their other ends at any three places of ,a1 ,b1 ,c1 respectively1 these being any three of the bodies outside the cluster of ,s#a and its twelve neighbours4 ,cut the wires across at any chosen positions in them2 and round off the cut ends1 just leaving contact between the rounded ends1 which we shall call ff1 gg1 hh4 ,do homogeneously for every other solid of the assemblage what we have done for ,s#a4 ,now bend the wires slightly so as to separate the pairs of points of contact1 taking care to keep them from touching any other bodies which they pass near on their courses between ,s#a and ,a1 ,b1 ,c respectively4 ,after having done this1 thoroughly rigidify all the wires thus altered4 ,we may now1 having three independent variables at our disposal1 so change the orientation of the molecules1 relatively to rows of the assemblage1 as to bring ff1 gg1 and hh again into contact4 ,we have thus six fresh points of ,s#a2 of which three are f1 g1 h2 and the other three are on the three extensions of ,s#a corresponding to the single extensions of ,a1 ,b1 ,c respectively1 which we have been making4 ,thus we have a real solution of the interesting geometrical problem3--,it is required so to form a homogeneous assemblage of solids of any arbitrarily given shape that each solid shall be touched by eighteen others4 ,this problem is determinate1 because the making of the three contacts ff1 gg1 hh1 uses up the three independent variables left at our disposal after we have first formed a homogeneous assemblage with twelve points of contact on each solid4 ,but our manner of finding a shape for each solid which can allow the solution of the problem to be real1 proves that the solution is essentially imaginary for every wholly convex shape4 #ce4 ,pausing for a moment longer to consider afresh the geometrical problem of putting arbitrarily given equal and similar solids together to make a homogeneous assemblage of which each member is touched by eighteen others1 we see immediately that it is determinate 7whether it has any real solution or not71 because when the shape of each body is given we have nine disposables for fixing the assemblage3 six for the character of the assemblage of the corresponding points1 and three for the orientation of each molecule relatively to rows of the assemblage of corresponding points4 ,these nine disposables are determined by the condition that each body has nine pairs of contacts with others4 ,suppose now a homogeneous assemblage of the given bodies1 in open order with no contacts1 to be arbitrarily made according to any nine arbitrarily chosen values for the six distances between a point of ,s#a and the corresponding points of its six pairs of nearest and next nearest neighbours 7 #a above71 and the three angles 7 #i above7 specifying the orientation of each body relatively to rows of the assemblage4 ,we may choose in any nine rows through ,s#a any nine pairs of bodies at equal distances on the two sides of ,s#a far or near1 for the eighteen bodies which are to be in contact with ,s#a4 ,hence there is an infinite number of solutions of the problem of which only a finite number can be real4 ,every solution of the problem of eighteen contacts is imaginary when the shape is wholly convex4 ,illustration3 ,,fig4 #ac4 #cf4 ,without for a moment imagining the molecules of matter to be hard solids of convex shape1 we may derive valuable lessons in the tactics of real crystals by studying the assemblage described in #bd and #be and represented in ,figs4 #ab and #ac4 ,i must for the present forego the very attractive subject of the tactics presented by faces not parallel to one or other of the four faces of the primitive tetrahedrons which we found in #bd1 and ask you only to think of the two sides of a plate of crystal parallel to any one of them1 that is to say1 an assemblage of such layers as those represented geometrically in ,fig4 #ab and shown in stereoscopic view in ,fig4 #ac4 ,if1 as is the case with the solids#aj photographed in ,fig4 #ac1 the under side of each solid is nearly plane but slightly convex1 and the top is somewhat sharply curved1 we have the kind of difference between the upper and under of the two parallel sides of the crystal which ,i have already described to you in #ba above4 ,in this case the assemblage is formed by letting the solids fall down from above and settle in the hollows to which they come most readily1 or which give them the stablest position4 ,it would1 we may suppose1 be the hollows p q r1 not p q r1 7,fig4 #ab7 that would be chosen2 and thus1 of the two formations described in #be1 we should have that in which the hollows above p q r are occupied by the comparatively flat under sides of the molecules of the layer above1 and the hollows below the apertures p q r by the comparatively sharp tops of the molecules of the layers below4 #cg4 ,for many cases of natural crystals of the wholly asymmetric character1 the true forces between the crystalline molecules will determine precisely the same tactics of crystallization as would be determined by the influence of gravity and fluid viscosity in the settlement from water1 of sand composed of uniform molecules of the wholly unsymmetrical convex shape represented in ,figs4 #ab and #ac4 ,thus we can readily believe that a real crystal which is growing by additions to the face seen in ,fig4 #ab1 would give layer after layer regularly as ,i have just described4 ,but if by some change of circumstances the plate1 already grown to a thickness of many layers in this way1 should come to have the side facing from us in the diagram exposed to the mother-liquor1 or mother-gas1 and begin to grow from that face1 the tactics might probably be that each molecule would find its resting-place with its most nearly plane side in the wider hollows under p q r1 instead of with its sharpest corner in the narrower and steeper hollows under p q r1 as are the molecules in the layer below that shown in the diagram in the first formation4 ,the result would be a compound crystal consisting of two parts1 of different crystalline quality1 cohering perfectly together on the two sides of an interfacial plane4 ,it seems probable that this double structure may be found in nature1 presented by crystals of the wholly unsymmetric class1 though it may not hitherto have been observed or described in crystallographic treatises4 ,illustration3 ,,fig4 #ad4 #ch4 ,this asymmetric double crystal becomes simply the well-known symmetrical #8twin-crystal'#aa in the particular case in which each of the constituent molecules is symmetrical on the two sides of a plane through it parallel to the plane of our diagrams1 and also on the two sides of some plane perpendicular to this plane4 ,we see1 in fact1 that in this case if we cut in two the double crystal by the plane of ,fig4 #ad1 and turn one part ideally through #ahj round the intersection of these two planes1 we bring it into perfect coincidence with the other part4 ,this we readily understand by looking at ,fig4 #ad1 in which the solid shown in outline may be either an egg-shaped figure of revolution1 or may be such a figure flattened by compression perpendicular to the plane of the diagram4 ,the most readily chosen and the most stable resting-places for the constituents of each successive layer might be the wider hollows p q r3 and therefore if1 from a single layer to begin with1 the assemblage were to grow by layer after layer added to it on each side1 it might probably grow as a twin-crystal4 ,but it might also be that the presence of a molecule in the wider hollow p q r on one side1 might render the occupation of the corresponding hollow on the other side by another molecule less probable1 or even impossible4 ,hence1 according to the configuration and the molecular forces of the particular crystalline molecule in natural crystallization1 there may be necessarily1 or almost necessarily1 the twin1 when growth proceeds simultaneously on the two sides3 or the twin growth may be impossible1 because the first occupation of the wider hollows on one side1 may compel the continuity of the crystalline quality throughout1 by leaving only the narrower hollows p q r free for occupation by molecules attaching themselves on the other side4 #ci4 ,or the character of the crystalline molecule may be such that when the assemblage grows by the addition of layer after layer on one side only1 with a not very strongly decided preference to the wider hollows p q r1 some change of circumstances may cause the molecules of one layer to place themselves in a hollow p q r4 ,the molecules in the next layer after this would find the hollows p q r occupied on the far side1 and would thus have a bias in favour of the hollows p q r4 ,thus layer after layer might be added1 constituting a twinned portion of the growth1 growing1 however1 with less strong security for continued homogeneousness than when the crystal was growing1 as at first1 by occupation of the wider hollows p q r4 ,a slight disturbance might again occur1 causing the molecules of a fresh layer to settle1 not in the narrow hollows p q r1 but in the wider hollows p q r1 notwithstanding the nearness of molecules already occupying the wider hollows on the other side4 ,disturbances such as these occurring irregularly during the growth of a crystal1 might produce a large number of successive twinnings at parallel planes with irregular intervals between them1 or a large number of twinnings in planes at equal intervals might be produced by some regular periodic disturbance occurring for a certain number of periods1 and then ceasing4 ,whether regular and periodic1 or irregular1 the tendency would be that the number of twinnings should be even1 and that after the disturbances cease the crystal should go on growing in the first manner1 because of the permanent bias in favour of the wider hollows p q r4 ,these changes of molecular tactics1 which we have been necessarily led to by the consideration of the fortuitous concourse of molecules1 are no doubt exemplified in a large variety of twinnings and counter-twinnings found in natural minerals4 ,in the artificial crystallization of chlorate of potash they are of frequent occurrence1 as is proved1 not only by the twinnings and counter-twinnings readily seen in the crystalline forms1 but also by the brilliant iridescence observed in many of the crystals found among a large multitude1 which was investigated scientifically by ,sir ,george ,stokes ten years ago1 and described in a communication to the ,royal ,society #8,on a remarkable phenomenon of crystalline reflection' 7,proc4 ,r4,s41 vol4 xxxviii1 #ahhe1 p4 #agd74 #dj4 ,a very interesting phenomenon1 presented by what was originally a clear homogeneous crystal of chlorate of potash1 and was altered by heating to about #bde-bdh ,cent41 which ,i am able to show you through the kindness of ,lord ,rayleigh1 and of its discoverer1 ,mr4 ,madan1 presents another very wonderful case of changing molecular tactics1 most instructive in respect of the molecular constitution of elastic solids4 ,when ,i hold this plate before you with the perpendicular to its plane inclined at #aj or more to your line of vision1 you see a tinsel-like appearance1 almost as bright as if it were a plate of polished silver1 on this little area1 which is a thin plate of chlorate of potash cemented for preservation between two pieces of glass2 and1 when ,i hold a light behind1 you see that the little plate is almost perfectly opaque like metal foil4 ,but now when ,i hold it nearly perpendicular to your line of vision the tinsel-like appearance is lost4 ,you can see clearly through the plate1 and you also see that very little light is reflected from it4 ,as a result both of ,mr4 ,madan's own investigations1 and further observations by himself1 ,lord ,rayleigh came to the conclusion that the almost total reflection of white light which you see is due to the reflection of light at many interfacial planes between successive layers of twinned and counter-twinned crystal of small irregular thicknesses1 and not to any splits or cavities or any other deviation from homogeneousness than that presented by homogeneous portions of oppositely twinned-crystals in thorough molecular contact at the interfaces4 #da4 ,when the primitive clear crystal was first heated very gradually by ,madan to near its melting-point 7#cei according to ,carnelly71 it remained clear1 and only acquired the tinsel appearance after it had cooled to about #bde or #bdhab4 ,rayleigh found that if a crystal thus altered was again and again heated it always lost the tinsel appearance1 and became perfectly clear at some temperature considerably below the melting-point1 and regained it at about the same temperature in cooling4 ,it seems1 therefore1 certain that at temperatures above #bdh1 and below the melting-point1 the molecules had so much of thermal motions as to keep them hovering about the positions of p q r1 p q r1 of our diagrams1 but not enough to do away with the rigidity of the solid2 and that when cooled below #bdh the molecules were allowed to settle in one or other of the two configurations1 but with little of bias for one in preference to the other4 ,it is certainly a very remarkable fact in ,natural ,history1 discovered by these observations1 that1 when the molecules come together to form a crystal out of the watery solution1 there should be so much more decided a bias in favour of continued homogeneousness of the assemblage than when1 by cooling1 they are allowed to settle from their agitations in a rigid1 but nearly melting1 solid4 #db4 ,but even in crystallization from watery solution of chlorate of potash the bias in favour of thorough homogeneousness is not in every contingency decisive4 ,in the first place1 beginning1 as the formation seems to begin1 from a single molecular plane layer such as that ideally shown in ,fig4 #ad1 it goes on1 not to make a homogeneous crystal on the two sides of this layer1 but probably always so as to form a twin-crystal on its two sides1 exactly as described in #ch1 and1 if so1 certainly for the reason there stated4 ,this is what ,madan calls the #8inveterate tendency to produce twins 7such as would assuredly drive a ,malthus to despair7#ac'2 and it is to this that he alludes as #8the inevitable twin-plate' in the passage from his paper given in the foot-note to #da above4 #dc4 ,in the second place1 ,i must tell you that many of the crystals produced from the watery solution by the ordinary process of slow evaporation and crystallization1 show twinnings and counter-twinnings at irregular intervals in the otherwise homogeneous crystal on either one or both sides of the main central twin-plane1 which henceforth1 for brevity1 ,i shall call 7adopting the hypothesis already explained1 which seems to me undoubtedly true7 the #8initial plane4' ,each twinning is followed1 ,i believe1 by a counter-twinning at a very short distance from it2 at all events ,lord ,rayleigh's observations#ad prove that the whole number of twinnings and counter-twinnings in a thin disturbed stratum of the crystal on one side of the main central twin-plane is generally1 perhaps always1 even2 so that1 except through some comparatively very small part or parts of the whole thickness1 the crystal on either side of the middle or initial plane is homogeneous4 ,this is exactly the generally regular growth which ,i have described to you 7 #ci7 as interrupted occasionally or accidentally by some unexplained disturbing cause1 but with an essential bias to the homogeneous continuance of the more easy or natural one of the two configurations4 #dd4 ,i have now great pleasure in showing you a most interesting collection of the iridescent crystals of chlorate of potash1 each carefully mounted for preservation between two glass plates1 which have been kindly lent to us for this evening by ,mr4 ,madan4 ,in ,march1 #ahed1 ,dr4 ,w4 ,bird ,herapath sent to ,prof4 ,stokes some crystals of chlorate of potash showing the brilliant and beautiful colours you now see1 and1 thirty years later1 ,prof4 ,e4 ,j4 ,mills recalled his attention to the subject by sending him #8a fine collection of splendidly coloured crystals of chlorate of potash of considerable size1 several of the plates having an area of a square inch or more1 and all of them being thick enough to handle without difficulty4' ,the consequence was that ,stokes made a searching examination into the character of the phenomenon1 and gave the short1 but splendidly interesting1 communication to the ,royal ,society of which ,i have already told you4 ,the existence of these beautifully coloured crystals had been well known to chemical manufacturers for a long time1 but it does not appear that any mention of them was to be found in any scientific journal or treatise prior to ,stokes' paper of #ahhe4 ,he found that the colour was due to twinnings and counter-twinnings in a very thin disturbed stratum of the crystal showing itself by a very fine line1 dark or glistening1 according to the direction of the incident light when a transverse section of the plate of crystal was examined in a microscope4 ,by comparison with a spore of lycopodium he estimated that the breadth of this line1 and therefore the thickness of the disturbed stratum of the crystal1 ranged somewhere about the one-thousandth of an inch4 ,he found that the stratum was visibly thicker in those crystals which showed red colour than in those which showed blue4 ,he concluded that #8the seat of the coloration is certainly a thin twinned stratum' 7that is to say1 a homogeneous portion of crystal between a twinning and a counter-twinning71 and found that #8a single twin-plane does not show anything of the kind4' #de4 ,a year or two later ,lord ,rayleigh entered on the subject with an exhaustive mathematical investigation of the reflection of light at a twin-plane of a crystal 7,philosophical ,magazine1 ,september1 #ahhh71 by the application of which1 in a second paper #8,on the remarkable phenomenon of ,crystalline ,reflection described by ,prof4 ,stokes1' published in the same number of the ,philosophical ,magazine1 he gave what seems certainly the true explanation of the results of ,sir ,george ,stokes' experimental analysis of these beautiful phenomena4 ,he came very decidedly to the conclusion that the selective quality of the iridescent portion of the crystal1 in virtue of which it reflects almost totally light nearly of one particular wave-length for one particular direction of incidence 7on which the brilliance of the coloration depends71 cannot be due to merely a single twin-stratum1 but that it essentially is due to a considerable number of parallel twin-strata at nearly equal distances4 ,the light reflected by this complex stratum is1 for any particular direction of incident and reflected ray1 chiefly that of which the wave-length is equal to twice the length of the period of the twinning and counter-twinning1 on a line drawn through the stratum in the direction of either the incident or the reflected ray4 #df4 ,it seems to me probable that each twinning is essentially followed closely by a counter-twinning4 ,probably three or four of these twin-strata might suffice to give colour2 but in any of the brilliant specimens as many as twenty or thirty1 or more1 might probably be necessary to give so nearly monochromatic light as was proved by ,stokes' prismatic analysis of the colours observed in many of his specimens4 ,the disturbed stratum of about a one-thousandth of an inch thickness1 seen by him in the microscope1 amply suffices for the #e1 #aj1 or #ajj half wave-lengths required by ,rayleigh's theory to account for perceptible or brilliant coloration4 ,but what can be the cause of any approach to regular periodicity in the structure sufficiently good to give the colours actually observed8 ,periodical motion of the mother-liquor relatively to the growing crystal might possibly account for it4 ,but ,lord ,rayleigh tells us that he tried rocking the pan containing the solution without result4 ,influence of light has been suggested1 and ,i believe tried1 also without result1 by several enquirers4 ,we know1 by the beautiful discovery of ,edmond ,becquerel1 of the prismatic colours photographed on a prepared silver plate by the solar spectrum1 that #8standing waves' 7that is to say1 vibrations with stationary nodes and stationary places of maximum vibration71 due to co-existence of incident and reflected waves1 do produce such a periodic structure as that which ,rayleigh's theory shows capable of giving a corresponding tint when illuminated by white light4 ,it is difficult1 therefore1 not to think that light may be effective in producing the periodic structure in the crystallization of chlorate of potash1 to which the iridescence is due4 ,still1 experimental evidence seems against this tempting theory1 and we must perforce be content with the question unanswered3--,what can be the cause of #e1 or #aj1 or #ajj pairs of twinning and counter-twinning following one another in the crystallization with sufficient regularity to give the colour3 and why1 if there are twinnings and counter-twinnings1 are they not at irregular intervals1 as those produced by ,madan's process1 and giving the observed white tinsel-like appearance with no coloration8 #dg4 ,and now ,i have sadly taxed your patience3 and ,i fear ,i have exhausted it and not exhausted my subject6 ,i feel ,i have not got halfway through what ,i hoped ,i might be able to put before you this evening regarding the molecular structure of crystals4 ,i particularly desired to speak to you of quartz crystal with its ternary symmetry and its chirality#ae2 and to have told you of the etching#af by hydrofluoric acid which1 as it were1 commences to unbuild the crystal by taking away molecule after molecule1 but not in the reverse order of the primary up-building2 and which thus reveals differences of tactics in the alternate faces of the six-sided pyramid which terminates at either end1 sometimes at both ends1 the six-sided prism constituting generally the main bulk of the crystal4 ,i must confine myself to giving you a geometrical symbol for the ternary symmetry of the prism and its terminal pyramid4 ,illustration3 ,,fig4 #ae4 #dh4 ,make an equilateral equiangular hexagonal prism1 with its diagonal from edge to edge ninety-five hundredths#ag of its length4 ,place a number of these close together1 so as to make up a hexagonal plane layer with its sides perpendicular to the sides of the constituent hexagonal prisms3 see ,fig4 #ae and imagine the semicircles replaced by their diameters4 ,you see in each side of the hexagonal assemblage1 edges of the constituent prisms1 and you see at each corner of the assemblage a face 7not an edge7 of one of the constituent prisms4 ,build up a hexagonal prismatic assemblage by placing layer after layer over it with the constituent prisms of each layer vertically over those in the layer below2 and finish the assemblage with a six-sided pyramid by building upon the upper end of the prism1 layer after layer of diminishing hexagonal groups1 each less by one circumferential row than the layer below it4 ,you thus have a crystal of precisely the shape of a symmetrical specimen of rock crystal1 with the faces of its terminal pyramid inclined at #ch #ac to the faces of the prism from which they spring4 ,but the assemblage thus constituted has #8senary' 7or six-rayed symmetry74 ,to reduce this to ternary symmetry1 cut a groove through the middle of each alternate face of the prismatic molecule1 making this groove in the first place parallel to the edges3 and add a corresponding projection1 or fillet1 to the middles of the other three faces1 so that two of the cylinders similarly oriented would fit together1 with the projecting fillet on one side of one of them entering the groove in the anti-corresponding side of the other4 ,the prismatic portion of the assemblage thus formed shows 7see ,fig4 #ae71 on its alternate edges1 faces of molecules with projections and faces of molecules with grooves2 and shows only orientational differences between alternate faces1 whether of the pyramid or of the prism4 ,having gone only so far from #8senary' symmetry1 we have exactly the triple1 or three-pair1 anti-symmetry required for the piezo-electricity of quartz investigated so admirably by the brothers ,curie#ah1 who found that a thin plate of quartz crystal cut from any position perpendicular to a pair of faces of a symmetrical crystal1 becomes positively electrified on one side and negatively on the other when pulled in a direction perpendicular to those faces4 ,but this assemblage has not the chiral piezo-electric quality discovered theoretically by ,voigt#ai1 and experimentally in quartz and in tourmaline by himself and ,riecke#bj1 nor the well-known optic chirality of quartz4 ,illustration3 ,,fig4 #af4 ,illustration3 ,,fig4 #ag4 #di4 ,change now the directions of the grooves and fillets to either of the oblique configurations shown in ,fig4 #af1 which ,i call right-handed1 because the directions of the projections are tangential to the threads of a three-thread right-handed screw1 and ,fig4 #ag 7left-handed74 ,the prisms with their grooves and fillets will still all fit together if they are all right-handed1 or all left-handed4 ,illustration3 ,,fig4 #ah4 ,fig4 #ah shows the upper side of a hexagonal layer of an assemblage thus composed of the right-handed molecule of ,fig4 #af4 ,fig4 #ae unchanged1 still represents a horizontal section through the centres of the molecules4 ,a prism built up of such layers1 and finished at each end with a pyramid according to the rule of #dh1 has all the qualities of ternary chiral symmetry required for the piezo-electricity of quartz2 for the orientational differences of the alternate pairs of prismatic faces2 for the absolute difference between the alternate pairs of faces of each pyramid which are shown in the etching by hydrofluoric acid2 for the merely orientational difference between the parallel faces of the two pyramids2 and for the well-known chiro-optic#ba property of quartz4 ,look at two contiguous faces ,a1 ,b of our geometrical model quartz crystal now before you1 with its axis vertical4 ,you will see a difference between them3 turn it upside down2 ,b will be undistinguishable from what ,a was1 and ,a will be undistinguishable from what ,b was4 ,look at the two terminal pyramids1 and you will find that the face above ,a and the face below ,b are identical in quality1 and that they differ from the face above ,b and below ,a4 ,this model is composed of the right-handed constituent molecules shown in ,fig4 #af4 ,it is so placed before you that the edge of the prismatic part of the assemblage nearest to you shows you filleted faces of the prismatic molecules4 ,you see two pyramidal faces2 the one to your right hand1 over ,b1 presents complicated projections and hollows at the corners of the constituent molecules2 and the pyramidal face next your left hand1 over ,a1 presents their unmodified corners4 ,but it will be the face next your left hand which will present the complex bristling corners1 and the face next your right hand that will present the simple corners1 if1 for the model before you1 you substitute a model composed of left-handed molecules such as those shown in ,fig4 #ag4 #ej4 ,to give all the qualities of symmetry and anti-symmetry of the pyro-electric and piezo-electric properties of tourmaline investigated theoretically by ,voigt#bb1 and experimentally by himself and ,friecke#bc1 make a hollow in one terminal face of each of our constituent prisms1 and a corresponding projection in its other terminal face4 #ea4 ,coming back to quartz1 we can now understand perfectly the two kinds of macling which are well known to mineralogists as being found in many natural specimens of the crystal1 and which ,i call respectively the orientational macling1 and the chiral macling4 ,in the orientational macling all the crystalline molecules are right-handed1 or all left-handed2 but through all of some part of the crystal1 each of our component hexagonal prisms is turned round its axis through #fj from the position it would have if the structure were homogeneous throughout4 ,in each of the two parts the structure is homogeneous1 and possesses all the electric and optic properties which any homogeneous portion of quartz crystal presents1 and the facial properties of natural uncut crystal1 shown in the etching by hydrofluoric acid2 but there is a discontinuity at the interface1 not generally plane1 between the two parts1 which in our geometrical model would be shown by non-fittings between the molecules on the two sides of the interface1 while all the contiguous molecules in one part1 and all the contiguous molecules in the other part1 fit into one another perfectly4 ,in chiral macling1 which is continually found in amethystine quartz1 and sometimes in ordinary clear quartz crystals1 some parts are composed of right-handed molecules1 and others of left-handed molecules4 ,it is not known whether1 in this chiral macling1 there is or there is not also the orientational macling on the two sides of each interface2 but we may say probably not2 because we know that the orientational macling occurs in nature without any chiral macling1 and because there does not seem reason to expect that chiral macling would imply orientational macling on the two sides of the same interface4 ,i would like to have spoken to you more of this most interesting subject2 and to have pointed out to you that some of the simplest and most natural suppositions we can make as to the chemical forces 7or electrical forces1 which probably means the same thing7 concerned in a single chemical molecule of quartz1 ,si,o#b1 and acting between it and similar neighbouring molecules1 would lead essentially to these molecules coming together in triplets1 each necessarily either right-handed or left-handed1 but with as much probability of one configuration as of the other3 and to have shown you that these triplets of silica #c7,si,o#b7 can form a crystalline molecule with all the properties of ternary chiral symmetry1 typified by our grooved hexagonal prisms1 and can build up a quartz crystal by the fortuitous concourse of atoms4 ,i should like also to have suggested and explained the possibility that a right-handed crystalline molecule thus formed may1 in natural circumstances of high temperature1 or even of great pressure1 become changed into a left-handed crystal1 or vice-versa4 ,my watch1 however1 warns me that ,i must not enter on this subject4 ,illustration3 ,,fig4 #ai4 #eb4 ,coming back to mere molecular tactics of crystals1 remark that our assemblage of rounded1 thoroughly scalene1 tetrahedrons1 shown in the stereoscopic picture 7 #cf1 ,fig4 #ac above71 essentially has chirality because each constituent tetrahedron1 if wholly scalene1 has chirality#bd4 ,i should like to have explained to you how a single or double homogeneous assemblage of points has essentially no chirality1 and how three assemblages of single points1 or a single assemblage of triplets of points1 can have chirality1 though a single triplet of points cannot have chirality4 ,i should like indeed to have brought somewhat thoroughly before you the geometrical theory of chirality2 and in illustration to have explained the conditions under which four points1 or two lines1 or a line and two points1 or a combination of point1 line and plane1 can have chirality3 and how a homogeneous assemblage of non-chiral objects can have chirality2 but in pity ,i forbear1 and ,i thank you for the extreme patience with which you have listened to me4 ,,footnotes3 #a ,see foot-note on #bb below4 #b ,the holes in the cylinders are bored obliquely1 as shown in ,fig4 #d1 which causes them to remain at any desired position on the cord and allows them to be freed to move up and down by slackening the cord for a moment4 #c #8,on the ,homogeneous ,division of ,space1' by ,lord ,kelvin1 ,royal ,society ,proceedings1 vol4 lv1 ,jan4 #ah1 #ahid4 #d ,similar curves are said to be parallel when the tangents to them at corresponding points are parallel4 #e ,see foot-note to #ab above4 #f #8,on the ,division of ,space with ,minimum ,partitional ,area1' ,philosophical ,magazine1 vol4 xxiv1 #ahhg1 p4 #ejb1 and ,acta ,mathematica of the same year4 #g ,a4 ,levy1 ,edinburgh ,philosophical ,journal1 ,april1 #ahbb2 ,whewell1 ,phil4 ,trans4 ,royal ,society1 #ahbe2 ,miller1 ,treatise on ,crystallography4 #h ,i call any geometrical figure1 or group of points1 chiral1 and say that it has chirality1 if its image in a plane mirror1 ideally realized1 cannot be brought to coincide with itself4 ,two equal and similar right hands are homochirally similar4 ,equal and similar right and left hands are heterochirally similar or #8allochirally' similar 7but heterochirally is better74 ,these are also called #8enantiomorphs1' after a usage introduced1 ,i believe1 by ,german writers4 ,any chiral object and its image in a plane mirror are heterochirally similar4 #i ,philosophical ,magazine1 vol4 xx1 #ahhe1 second half year1 p4 #dfi1 and ,british ,association ,report1 #ahhe1 ,aberdeen1 p4 #hif4 #aj ,the solids of the photograph are castings in fine plaster of ,paris from a scalene tetrahedron of paraffin wax1 with its corners and edges rounded1 used as a pattern4 #aa #8,a twin-crystal is composed of two crystals joined together in such a manner that one would come into the position of the other by revolving through two right angles round an axis which is perpendicular to a plane which either is1 or may be1 a face of either crystal4 ,the axis will be called the twin-axis1 and the plane to which it is perpendicular the twin-plane4' ,miller's ,treatise on ,crystallography1 p4 #ajc4 ,in the text the word #8twin-plane1' quoted from the writings of ,stokes and ,rayleigh1 is used to signify the plane common to the two crystals in each of the cases referred to3 and not the plane perpendicular to this plane1 in which one part of the crystal must be rotated to bring it into coincidence with the other1 and which is the twin-plane as defined by ,miller4 #ab #8,a clear transparent crystal of potassium chlorate1 from which the inevitable twin-plate had been ground away so as to reduce it to a single crystal film about #a mm4 in thickness1 was placed between pieces of mica and laid on a thick iron plate4 ,about #c cm4 from it was laid a small bit of potassium chlorate1 and the heat of a ,bunsen burner was applied below this latter1 so as to obtain an indication when the temperature of the plate was approaching the fusing-point of the substance 7#cei ,c according to ,prof4 ,carnelly74 ,the crystal plate was carefully watched during the heating1 but no depreciation took place1 and no visible alteration was observed1 up to the point at which the small sentinel crystal immediately over the burner began to fuse4 ,the lamp was now withdrawn1 and when the temperature had sunk a few degrees a remarkable change spread quickly and quietly over the crystal plate1 causing it to reflect light almost as brilliantly as if a film of silver had been deposited upon it4 ,no further alteration occurred during the cooling2 and the plate1 after being ground and polished on both sides1 was mounted with ,canada balsam between glass plates for examination4 ,many crystals have been similarly treated with precisely similar results2 and the temperature at which the change takes place1 has been determined to lie between #bde and #bdh1 by heating the plates upon a bath of melted tin in which a thermometer was immersed4 ,with single crystal plates no decrepitation has ever been observed1 while with the ordinary twinned-plates it always occurs more or less violently1 each fragment showing the brilliant reflective power above noticed4'--,nature1 ,may #bj1 #ahhf4 #ac ,nature1 ,may #bj1 #ahhf4 #ad ,philosophical ,magazine1 #ahhh1 second half year1 p4 #bfj4 #ae ,see foot-note to #bb above4 #af ,widmansttten1 #ahjg4 ,leydolt 7#ahee1 ,wien4 ,akad4 ,ber4 #ae1 #ei1 ,t4 #i1 #aj4 ,baumhauer1 ,pogg4 ,ann4 #ach1 #efc 7#ahfi72 #adj1 #bga2 #adb1 #cbd2 #ade1 #dfj2 #aej1 #fai47 ,for an account of these investigations1 see ,mallard1 ,trait de ,crystallographie 7,paris1 #ahhd71 ,tome ,,ii1 chapitre xvi4 #ag ,more exactly 4#iebe1 being #c_/d cot #ch #ac2 see p4 #ec4 #ah ,j4 and ,p4 ,curie and ,c4 ,friedel1 ,comptes ,rendus1 #ahhb1 #ahhc1 #ahhf1 #ahib4 #ai ,allgemeine ,theorie der pizo- und pyroelectrischen ,erscheinungen an ,krystallen4 ,w4 ,voigt1 ,knigl4 ,gesellschaft der ,wissenschaften zu ,gttingen1 ,august #b1 #ahij4 #bj ,wiedemann1 ,annalen1 #ahib1 xlv1 p4 #ibc4 #ba ,generally miscalled #8rotational4' #bb ,see foot-note 7#b7 to p4 #ed above4 #bc ,see foot-note 7#c7 to p4 #ed above4 #bd ,see foot-note to #bb above4 ,,the ,,end ,oxford ,,printed ,,at ,,the ,,clarendon ,,press ,,by ,,horace ,,hart1 ,,printer ,,to ,,the ,,university