,transcriber's ,notes3 ,underscores 80 before and after a word or phrase indicate italics in the original text4 ,equal signs 80 before and after a word or phrase indicate bold in the original text4 ,small capitals have been converted to ,,solid capitals4 ,typographical errors have been silently corrected4 ,vol4 ,i4 ,,october1 #ahib ,no4 #b4 ,,the ,,kansas ,,university ,,quarterly ,,contents ,,unicursal ,,curves ,,by ,,method ,,of ,,inversion1 ,h4 ,b4 ,newson ,,foreign ,,settlements ,,in ,,kansas1 ,w4 ,h4 ,carruth ,,the ,,great ,,spirit ,,spring ,,mound1 ,r4 ,h4 ,s4 ,bailey ,,on ,,pascal',s ,,lima,,on ,,and ,,the ,,cardioid1 ,h4 ,c4 ,riggs ,,dialect ,,word-,,list1 ,w4 ,h4 ,carruth ,,published ,,by ,,the ,,university ,,lawrence1 ,,kansas ,price of this number1 #ej cents ,entered at the ,post-office in ,lawrence as ,second-class matter4 ,,committee ,,of ,,publication ,e4 ,h4 ,s4 ,,bailey ,f4 ,w4 ,,blackmar ,w4 ,h4 ,,carruth ,c4 ,g4 ,,dunlap ,e4 ,,miller ,s4 ,w4 ,,williston ,v4 ,l4 ,,kellogg1 ,,managing ,,editor ,,journal ,,publishing ,,house ,,lawrence1 ,,kansas #ahib ,,kansas ,,university ,,quarterly4 ,,vol4 ,i4 ,,october1 #ahib4 ,,no4 #b4 ,unicursal ,curves by ,method of ,inversion4 ,,by ,,henry ,,byron ,,newson4 ,this paper contains a summary of the work done during the last school year by my class in ,modern ,geometry4 ,since many of the results were suggested or entirely wrought out by class-room discussion1 it becomes practically impossible to assign to each member of the class his separate portion4 ,many of the results were contributed by ,messrs4 ,m4 ,e4 ,rice1 ,a4 ,l4 ,candy1 ,h4 ,c4 ,riggs1 and ,miss ,annie ,l4 ,mac,kinnon4 ,the reader who is not familiar with the method of ,geometric ,inversion should read ,townsend's ,modern ,geometry1 chapters ,,ix and ,,xxiv2 or a recent monograph entitled1 8,das ,princep der ,reziproken ,radien10 by ,c4 ,wolff1 of ,erlangen4 ,when a conic is inverted from a point on the curve1 the inverse curve is a nodal1 circular cubic4 ,this is shown analytically as follows3 let the equation of the conic be written ax #b;hxy by #b;gx #b;fy #j2 which shows that the origin is a point on the curve4 ,substituting for x y x and y ------------------ and ---------------- 1 x y 7x y7 we have as the equation of the inverse curve ax #b;hxy by #b7;gx fy77x y7 #j4 ,the terms of the second degree show that the origin is a double point on the cubic2 and is a crunode1 acnode1 or cusp1 according as the conic is a hyperbola1 ellipse1 or parabola4 ,the terms of the third degree break up into three linear factors1 viz3 gx fy1 x iy1 and x - iy1 which are the equations of the three lines joining the origin to the three points where the line at infinity cuts the cubic2 thus showing that the cubic passes through the imaginary circular points at infinity4 ,since the above transformation is rational1 it follows that there is a 7#a1 #a7 correspondence between the conic and the cubic4 ,this fact is also evident from the nature of the method of inversion4 ,the cubic has its maximum number of double points1 viz3 one2 and hence is unicursal4 ,this unicursal circular cubic may be projected into the most general form of unicursal cubic2 the cuspidal variety1 however1 always remaining cuspidal4 ,by applying the method of inversion to many of the well known theorems of conics1 new theorems are obtained for unicursal1 circular cubics4 ,if one of these new theorems states a projective property1 it may at once by the method of projection be extended to all unicursal cubics4 ,examples will be given below4 ,the following method of generating a unicursal cubic is often useful4 ,given two projective pencils of rays having their vertices at ,a and ,b2 the locus of the intersection of corresponding rays is a conic through ,a and ,b4 ,invert the whole system from ,a4 ,the pencil through ,a remains as a whole unchanged1 while the pencil through ,b inverts into a system of co-axial circles through ,a and ,b1 and the generated conic becomes a circular cubic through ,a and ,b1 having a node at ,a4 ,now project the whole figure and we have the following3--given a system of conics through four fixed points and a pencil of rays projective with it and having its vertex at one of the fixed points1 the locus of the intersection of corresponding elements of the two systems is a unicursal cubic1 having its node at the vertex of the pencil1 and passing through the three other fixed points4 ,unicursal cubics are divisible into two distinct varieties1 nodal and cuspidal4 ,the nodal variety is a curve of the fourth class and has three points of inflection1 one of which is always real4 ,the cuspidal variety is of the third class and has one point of inflection 7,salmon1 ,h4 ,p4 ,c41 ,art4 #adg74 ,each of these varieties forms a group projective within itself2 that is to say1 any nodal cubic may be projected into every other possible nodal cubic1 and the same is true with regard to the cuspidal4 ,but a nodal cubic can not be projected into a cuspidal and vice versa4 ,in applying this method of investigation to the various forms of unicursal cubics and quartics1 only a limited number of theorems are given in each case4 ,it will be at once evident that many more theorems might be added1 but enough are given in each case to illustrate the method and show the range of its application4 ,it is not necessary to work out all the details1 as this paper is intended to be suggestive rather than exhaustive4 ,,nodal ,,cubics4 ,if an ellipse be inverted from one of its vertices1 the inverse curve is symmetrical with respect to the axis2 it has one point of inflection at infinity and the asymptote is an inflectional tangent4 ,this asymptote is the inverse of the circle of curvature at the vertex4 ,the cubic has two other points of inflection situated symmetrically with respect to the axis4 ,hence the three points of inflection lie on a right line1 a projective theorem which is consequently true of all nodal cubics4 ,the axis is evidently the harmonic polar of the point of inflection at infinity4 ,since the axis bisects the angle between the tangents at the node1 it follows that the line joining a point of inflection to the node1 the two tangents at the node1 and the harmonic polar of the point of inflection1 form a harmonic pencil4 ,there are three such lines1 one to each node1 and three harmonic polars2 these form a pencil in involution1 the tangents at the node being the foci4 ,since the asymptote is perpendicular to the axis1 we have by projection the following theorem3--through a point of inflection ,i1 draw any line cutting the cubic in ,b and ,c4 ,through ,p the point of intersection of the harmonic polar and inflectional tangent of ,i1 draw two lines to ,b and ,c4 ,the four lines meeting in ,p form a harmonic pencil4 ,the point of contact of the tangent from ,i to the cubic is on the harmonic polar of ,i4 ,any two inflectional tangents meet on the harmonic polar of the third point of inflection4 ,the locus of the foot of the perpendicular from the focus of a conic on a tangent is the auxiliary circle4 ,inverting from the vertex1 there are two points1 ,a and ,b1 on the axis of the curve1 such that if a circle be drawn through one of them and the node1 cutting at right angles a tangent circle through the node1 their point of intersection will be on the tangent to the curve where it is cut by the axis4 ,projecting3--through a point of inflection ,i of a nodal cubic draw a line cutting the cubic in ,p and ,q2 there are two determinate points on the harmonic polar of ,i1 which have the following property3--draw a conic through ,p1 ,q1 and the node touching the cubic2 draw another conic through one of these points1 ,p1 ,q1 and the node cutting the former1 so that their tangents at their point of intersection1 together with the lines from it to ,p and ,q form a harmonic pencil2 the locus of such a point of intersection is the tangent from ,i to the cubic4 ,if three conics circumscribe the same quadrilateral1 the common tangent to any two is cut harmonically by the third4 ,inverting from one of the vertices of the quadrilateral3 if three nodal1 circular cubics have a common double point and pass through three other fixed points1 the common tangent circle through the common node to any two of the cubics is cut harmonically by the third2 i4 e41 so that the pencil from the node to the two points of intersection and the points of contact is harmonic4 ,projecting this3--given three nodal cubics having a common node and passing through five other fixed points2 let a conic be passed through the common node and two of the fixed points1 touching two of the cubics4 ,the pencil from the common node to the points of contact and the point where the conic cuts the third cubic is harmonic4 ,the following theorem may be proved in similar manner3--given a system of cubics having a common node and passing through five other fixed points2 let a conic be drawn through the common node and two of the fixed points2 the lines drawn from the points where it cuts the cubics to the common node form a pencil in involution4 ,a variable chord drawn through a fixed point ,p to a conic subtends a pencil in involution at any point ,o on the conic4 ,inverting from ,o3--a system of circles through the double point of a nodal circular cubic and any other fixed point ,p1 is cut by the cubic in pairs of points which determine at the node a pencil in involution4 ,projecting3--a system of conics through the node of a unicursal cubic1 two fixed points on the curve1 and any fourth fixed point1 is cut by the cubic in pairs of points which determine at the node a pencil in involution4 ,we give another proof of the theorem that the three points of inflection of a nodal cubic lie on a right line4 ,this is easily shown by inversion and is a beautiful example of the method4 ,there are three points on a conic whose osculating circles pass through a given point on the conic2 these three points lie on a circle passing through the given point4#a 7,salmon's ,conics1 ,art4 #bdd1 ,ex4 #e47 ,by inverting from the given point and then projecting1 we readily see that there are three points of inflection on a nodal cubic which lie on a right line4 ,if the above conic be an ellipse1 the three osculating circles are all real2 but if it be a hyperbola1 one only is real4 ,hence an acnodal cubic has three real points of inflection1 while a crunodal one has one real and two imaginary4 ,the reciprocals of many of the theorems of this section are of interest and will be given under ,quartics4 ,,cuspidal ,,cubics4#b ,inverting the parabola from its vertex we obtain the ,cissoid of ,diocles4 ,the focus of the parabola inverts into a point on the cuspidal tangent which ,i shall call the focus of the cissoid4 ,the circle of curvature at the vertex of the parabola inverts into the asymptote of the cissoid4 ,this asymptote is also plainly the inflectional tangent1 and the point at infinity is the point of inflection4 ,the directrix of the parabola inverts into a circle through the cusp of the cissoid having the cuspidal tangent for a diameter4 ,hall calls this the directrix circle4 ,the double ordinate of the parabola which is tangent to the circle of curvature of the vertex inverts into the circle usually called the base circle of the cissoid4#c ,the cissoid may fairly be called the simplest form of the cuspidal cubic4 ,its projection and polar reciprocal are both cuspidal cubics4 ,i shall now deduce from the parabola a few simple propositions for the cissoid1 and then extend them to all cuspidal cubics4 7#a7 ,it is known that the locus of the intersection of tangents to the parabola which are at right angles to one another1 is the directrix4 ,inverting3--the locus of the intersection of tangent circles to the cissoid through the cusp and at right angles to each other is the directrix circle4 7#b7 ,for the parabola1 two right lines ,o ,p and ,o ,q1 are drawn through the vertex of the parabola at right angles to one another1 meeting the curve in ,p and ,q2 the line ,p ,q cuts the axis at a fixed point1 whose abscissa is equal to its ordinate4 ,inverting3--two right lines1 ,o ,p and ,o ,q1 are drawn at right angles to one another through the cusp of the cissoid1 meeting the curve in ,p and ,q2 the circle ,o ,p ,q passes through the intersection of the axis and asymptote4 7#c7 ,if the normals at the points ,p1 ,o1 ,r1 of a parabola meet at a point1 the circle through ,p ,o ,r will pass through the vertex4 ,inverting3--through a fixed point and the cusp of a cissoid1 three and only three circles can be passed1 cutting the cissoid at right angles2 these three points of intersection are collinear4 ,from the geometry of the cissoid we see that if any line be drawn parallel to the asymptote1 cutting the curve in two points1 ,b and ,c1 the segment ,b ,c is bisected by the axis4 ,hence1 projecting the curve we have the following theorem3--any line drawn through the point of inflection is cut harmonically by the point of inflection1 the curve1 and the cuspidal tangent4 ,thus the cuspidal tangent is the harmonic polar of the point of inflection4 ,the polar reciprocal of this last theorem reads as follows3--if from any point on the cuspidal tangent the two other tangent lines be drawn to the curve1 and a line to the point of inflection1 these four lines form a harmonic pencil4 ,these are fundamental propositions in the theory of cuspidal cubics4 7#d7 ,projecting proposition 7#a7 above1 we have the generalized theorem3--through the point of inflection draw any line cutting the cubic in ,b and ,c2 through ,b1 ,c1 and the cusp draw two conics tangent to the cubic1 and intersecting in a fourth point such that the two tangents to the conics at their point of intersection1 together with the two lines from it to ,b and ,c1 form a harmonic pencil2 the locus of all such intersections is a conic through ,b1 ,c1 and the cusp having the point of inflection and the cuspidal tangent for pole and polar4 7#e7 ,reciprocating 7#d7 we have3--through any point on the cuspidal tangent draw the two other tangents1 ,b and ,c1 to the cubic4 ,touching ,b1 ,c1 and the inflectional tangent draw two conics1 such that the points of contact of their common tangent1 together with the points where their common tangent cuts the tangents ,b and ,c1 form a harmonic range2 the envelope of such common tangents is a conic having the cuspidal tangent and the point of inflection for polar and pole4 7#f7 ,projecting 7#b7 we obtain the following3--through the point of inflection draw any line cutting the curve in ,b and ,c2 take any other two points on the cubic such that the pencil from the cusp1 ,o1 ,o 7,b ,p ,c ,q7 is harmonic2 the conic passing through ,o ,b ,p ,c ,q will pass through the intersection of the cuspidal and inflectional tangents4 7#g7 ,reciprocating 7#f73--;from any point on the cuspidal tangent draw two other tangents1 ,b and ,c1 to the cubic2 take any two other tangents1 ,p and ,q1 such that the range cut from the inflectional tangent by ,b1 ,c1 ,p1 ,q1 is harmonic2 the conic touching ,b1 ,c1 ,p1 ,q1 and the inflectional tangent will also touch the line joining the point of inflection and the cusp4 7#h7 ,projecting 7#c73--through the point of inflection draw any line cutting the cubic in ,b and ,c2 through the cusp ,o and the points ,b and ,c on the cubic and any other fixed point ,p1 three1 and only three1 conics can be passed1 such that the tangent to the conic and cubic at their remaining point of intersection1 together with the lines from it to ,b and ,c1 form a harmonic pencil2 these three points of intersection are collinear4 ,,systems ,,of ,,cubics ,,through ,,nine ,,points4 ,let ,u and ,v be the equations of two given cubics1 then ,u k,v is the equation of a system of cubics through their nine points of intersection4 ,twelve cubics of this system are unicursal1 and the twelve nodes are called the twelve critic centres of the system4 7,see ,salmon's ,h4 ,p4 ,c41 ,art4 #aij47 ,let the equation of the system be written briefly a ka 7b kb7 x 7c kc7 y u u #j2 one1 and only one1 value of k makes the absolute term vanish2 hence one1 and only one1 curve of the system passes through the origin1 which may be any point in the plane4 ,make the equation of the system homogeneous by means of z1 and differentiate twice with respect to z2 we obtain thus the equations of the polar conics and polar lines of the origin with respect to the system4 ,the polar conics of the origin are given by #c7;a ka7 #b 7b kb7 x 7c kc7y u #j2 thus showing that the polar conics of any point1 with respect to the system of cubics1 form a system through four points4 ,the polar lines of the origin are given by #c7;a ka7 7b kb7 x 7c kc7y #j1 which represents a pencil of lines through a point4 ,suppose now the origin to be at one of the critic centres2 then for a particular value1 k1 all terms lower than the second degree must vanish1 so that a b c #j4 a b c ,the factors of the terms of u1 which involves k1 represent the tangents at the double point to the nodal cubic1 and also the polar conic of the origin with respect to this nodal cubic4 ,hence a critic centre is at one of the vertices of the self-polar triangle of its system of polar conics4 ,the opposite side of this triangle is the common polar line of the critic centre with respect to its system of polar conics1 and hence it is also the common polar line of the critic centre with respect to the system of cubics4 ,the four basal points of the system of polar conics lie two and two upon the tangents at the double point of the nodal cubic4 ,when the origin is taken at one of the nine basal points of the system of cubics1 a and a both vanish4 ,hence it is readily seen that a basal point of a system of cubics is also a basal point of its system of polar conics and the vertex of its pencil of polar lines4 ,suppose two of the basal points of the system of cubics to coincide1 then every cubic of the system1 in order to pass through two coincident points1 must touch a common tangent at a fixed point4 ,the common tangent is the common polar of its point of contact1 both with respect to the system of cubics and to its system of polar conics4 ,hence the union of two basal points gives rise to a critic centre4 ,the self-polar triangle of its system of polar conics here reduces to a limited portion of the common tangent4 ,this line is not a tangent to the nodal cubic1 but only passes through its double point4 ,suppose three of the basal points of a system of cubics to coincide1 such a point will then be a point of inflection on each cubic of the system4 ,for1 in the last case1 if a line be drawn from the point of contact of the common tangent to a third basal point of the system1 such a line will be a common chord of the system of cubics4 ,suppose1 now1 this third basal point be moved along the curves until it coincides with the other two2 then the common chord becomes a common tangent1 which cuts every cubic of the system in three coincident points1 and hence is a common inflectional tangent4 ,since the polar conic of a point of inflection on a cubic consists of the inflectional tangent and the harmonic polar of the point1 and since the polar conics of a fixed point with respect to a system of cubics pass through four fixed points1 it follows that in a system of cubics having a common point of inflection and a common inflectional tangent the harmonic polars of the common point of inflection meet in a point4 ,since the common inflectional tangent is the common polar line of the common point of inflection1 it follows that such a point is a critic centre of the system of cubics4 ,one cubic of the system then has a node at the common point of inflection of the system1 and forms an exception4 ,the line which is the common inflectional tangent to the other cubics of the system cuts this also in three points1 but is one of the tangents at the double point2 the other tangent at the double point goes through the vertex of the pencil of harmonic polars4 ,it is evident that the nine basal points of a system of conics may unite into three groups of three each4 ,the cubics will then all have three common points of inflection1 and at these points three common inflectional tangents4 ,these three points all lie on a line4 ,when four basal points of the system of cubics coincide1 such a point is a double point on every cubic of the system4 ,this is easily shown as follows1 using the method of inversion4 ,let a system of conics through four points be inverted from one of the four points4 ,the system of conics inverts into a system of cubics1 having a common node and passing through three other finite fixed points and the two circular points at infinity4 ,since the common node counts as four points of intersection1 it follows that any two cubics of the system1 and hence all of them1 intersect in nine points4 ,this system can be projected into a system having a common double point and passing through any five other fixed points4 ,a number of theorems concerning the system of cubics can easily be inferred from known theorems concerning the system of conics4 ,since two conics of the system are parabolas1 it follows that two cubics of the system are cuspidal4 ,since three conics of the system break up into pairs of right lines1 it follows that three cubics of the system break up into a right line and a conic4 ,each right line and its corresponding conic intersect in the common double point4 ,the line at infinity cuts the system of conics in pairs of points in involution1 the points of contact of the two parabolas of the system being the foci2 it follows on inversion that the pairs of tangents to the cubics at their common node form a pencil in involution1 the two cuspidal tangents being the foci4 ,if the four basal points of the system of conics lie on a circle1 this circle inverts into a right line1 and one cubic then consists of this right line and the lines joining the centre of inversion to the circular points at infinity4 ,this theorem may be stated for the system of cubics as follows3 if the conic determined by the five basal points of the system of cubics 7not counting the common double points71 break up into right lines1 the line passing through three of the five points1 together with the lines joining the other two points to the common node1 constitute a cubic of the system4 ,if three of the four basal points of the system of conics lie on a line1 the conics consist of this line and a pencil of lines through the fourth basal point4 ,inverting from this fourth point and then projecting1 we have a system of cubics consisting of a pencil of lines and a conic through the vertex and the four other fixed points4 ,hence1 when the five fixed points of such a system of cubics lie on a conic through the common node1 this conic is a part of every cubic of the system4 ,if we invert the above system of conics from one of the three points on the right line1 and then project1 we obtain a system of cubics which consists of a system of conics through four fixed points1 and a fixed right line through one of these four points4 ,hence1 if two of the five basal points of such a system of cubics be on a line through the common node1 this line is a part of every cubic of the system4 ,if a system of conics having one basal point at infinity be inverted from one of the remaining basal points1 this point at infinity inverts to the center of inversion1 and we obtain a system of cubics having five coincident basal points and hence passing through only four others4 ,the system of cubics is now so arranged that one tangent at their common double point is common to all4 ,only one cubic of the system is cuspidal4 ,as before three cubics break up into a right line and conic4 ,if two of three basal points of the system of conics be at infinity1 the system of cubics obtained by projection and inversion has six coincident basal points and hence only three others4 ,this system has both tangents at the common node common to all cubics of the system4 ,if the two basal points at infinity in the system of conics be coincident1 all the conics are parabolas1 and hence all the cubics of the system are cuspidal and have a common cuspidal tangent4 ,if three of the basal points of the system of conics be at infinity1 the conics consist of the line at infinity and a pencil of lines through the finite basal point4 ,inverting from the latter1 we obtain a system of cubics with seven coincident basal points4 ,this system is made up of a pencil of lines meeting in the seven coincident basal points together with the two lines joining this to the other two basal points of the system4 ,these two lines are part of every cubic of the system4 ,if one of the remaining basal points be moved up to join the seven coincident ones1 one of these fixed lines becomes indeterminate1 and the system of cubics through eight coincident points consists of a fixed line through the eight coincident points and the ninth fixed point together with any two lines of the pencil through the eight points4 ,if the nine basal points coincide1 any three lines through it form a cubic of the system4 ,,unicursal ,,quartics4 ,the inverse of a conic from any point not on the curve is a nodal bicircular quartic4 ,this is shown by inverting the general equation of the conic ax #b;hxy by #b;gx #b;fy c #j2 x y by substituting for x and y1 ------------------ and -------------------- 1 7x y7 7x y7 we get the equation ax #b;hxy by #b7;gx fy77x y7 c7x y7 #j4 ,the origin is evidently a double point on the curve1 and is a crunode1 acnode1 or cusp according as the conic is a hyperbola1 ellipse1 or parabola4 ,the factors of the terms of the fourth degree1 viz3 7x iy7 7x iy7 7x - iy7 7x - iy71 show that the two imaginary circular points at infinity are double points on the quartic1 which is thus trinodal4 ,hence this nodal1 bicircular quartic can be projected into the most general form of the trinodal quartic4 ,trinodal quartics are unicursal4 ,if the conic which we invert be a parabola1 the quartic has two nodes and one cusp4 ,if the conic be inverted from a focus1 the quartic has the two circular points at infinity for cusps4 ,this is best shown analytically as follows3 let the equation of the conic1 origin being at the focus1 be written x y #b;aex b ------ ------ -------- - ------ #j4 a b a a ,inverting this we have x y #b;aex7x y7 b7x y7 ------ ------ ------------------------------ - -------------------------------- #j4 a b a a ,now transform this equation so that the lines joining the origin to the circular points at infinity shall be the axes of reference4 ,to do this let x iy x and x - iy y2 x y x - y x ---------------- and y ---------------- 4 #b #b;i4 ,making these substitutions and reducing we have 7dropping the subscripts71 7x #bxy y7 7x - #bxy y7 #d;aexy7x y7 bxy ---------------------------------- - ---------------------------------- - ------------------------ - ------------------ #j4 a b a a ,making this equation homogeneous by means of z1 we have 7x #bxy y7 7x - #bxy y7 #d;aexyz7x y7 bxy z -------------------------------- - -------------------------------- - -------------------------- - ------------------ a b a a #d;aexyz7x y7 bxy - -------------------------- - ------------------ #j1 a a which is the equation of the quartic referred to the triangle formed by the three nodes4 ,we are now able to determine the nature of the node at the vertex 7y1 z74 ,factor x out of all the terms which contain it2 and arrange thus3 z z #d;aeyz by yz yz #b;aeyz x ------ - ------ - ---------- - ------------ #bx -------- -------- - -------------- a b a a a b a yz yz ------------ - ------------ #j4 a b ,the quantity which multiplies x represents the two tangents at the double point 7y1 z72 but this quantity is a perfect square and hence we have a cusp4 ,in this way the point 7x1 z7 may be shown to be a cusp4 ,lastly1 when a parabola is inverted from the focus1 we obtain a tricuspidal quartic4 ,the trinodal quartic can be generated in a manner analogous to that shown for the nodal cubic4 ,let two projective pencils of rays have their vertices at ,a and ,b1 the locus of intersection of corresponding rays is a conic through ,a and ,b4 ,invert from any point ,o in the plane1 and we obtain two systems of co-axial circles1 ,o ,a being the axis of one and ,o ,b of the other4 ,the locus of intersection of corresponding circles is a bicircular quartic having a node at ,o4 ,projecting the whole figure we have the following theorem3--two projective systems of conics through ,o ,p ,q ,a and ,o ,p ,q ,b generate by their corresponding intersections a trinodal quartic having its nodes at ,o1 ,p1 and ,q1 and passing through ,a and ,b4 ,it is evident that the quartic generated in this way may have three nodes1 one node and two cusps1 two nodes and one cusp1 or three cusps1 depending upon the nature of the conic inverted and the centre of inversion4 ,making this the basis of classification we thus distinguish four varieties of unicursal quartics4 ,to these must be added a fifth variety1 viz3 the quartic with a triple point4 ,each of these varieties will be considered separately4 ,the method of treating unicursal quartics given in this and the next four sections is in some respects similar to that suggested by ,cayley in ,salmon's ,higher ,plane ,curves4 ,but the method here sketched out is very different in its point of view and much wider in its application1 yielding a multitude of new theorems not suggested by ,cayley's method4 ,,trinodal ,,quartics4 ,the quartic with three double points is a curve of the sixth class having four double tangents and six cusps 7,salmon's ,h4 ,p4 ,c4 ,art4 #bdc74 ,hence its reciprocal is of the sixth degree with four double points1 six cusps1 three double tangents1 and no points of inflection4 ,the locus of intersection of tangents to a conic at right angles to one another is a circle4 ,inverting3--the locus of intersection of circles through the node and tangent a nodal1 bicircular quartic and at right angles to one another is a circle4 ,projecting3--through the three nodes of a quartic draw two conics1 each touching the quartic and intersecting so that the two tangents to the conics at their point of intersection1 together with the lines from it to two of the nodes1 form a harmonic pencil2 the locus of all such intersections is a conic through these two nodes4 ,whenever the two tangents to the quartic from the third node1 together with the lines from it to the other two nodes1 form a harmonic pencil1 this last conic breaks up into two right lines4 ,any chord of a conic through ,o is cut harmonically by the conic and the polar of ,o4 ,inverting from ,o and projecting3--from one of the nodes of a trinodal quartic draw the two tangents to the quartic 7not tangents at the node72 draw the conic through these two points of contact and the three nodes2 any line through the first mentioned node is cut harmonically by this conic1 the quartic and the line joining the other two nodes4 ,if a triangle circumscribe a conic1 the three lines from the angular points of the triangle to the points of contact of the opposite sides intersect in a point4 ,inverting and projecting3--through the three nodes of a quartic draw three conics touching the quartic2 through the point of intersection of two of these conics1 the point of contact of the third1 and the three nodes draw a conic2 three such conics can be drawn and they pass through a fixed point4 ,the eight points of contact of two conics with their four common tangents lie on a conic1 which is the locus of a point1 the pairs of tangents from which to the two given conics form a harmonic pencil4 ,inverting and projecting3--two connodal trinodal quartics have four common tangent conics through the three nodes2 their eight points of contact lie on another connodal trinodal quartic2 if from any point on the last quartic four conics be drawn through the nodes and tangent in pairs to the first quartics1 any line through a node is cut harmonically by these four conics4 ,the eight common tangents to two conics at their common points all touch a conic4 ,inverting and projecting3--two connodal trinodal quartics intersect in four other points2 eight conics can be drawn through the three nodes tangent to the quartics at these points of intersection2 these eight conics all touch another connodal trinodal quartic4 ,a series of conics through four fixed points is cut by any transversal in a range of points in involution4 ,inverting and projecting3--a series of connodal trinodal quartics can be passed through four other fixed points2 any conic through the three nodes cuts the series of quartics in pairs of points which determine at a node a pencil in involution4 ,the conic touches two of the quartics and the lines to the points of contact are the foci of the pencil4 ,if the sides of two triangles touch a given conic1 their six angular points will lie on another conic4 ,inverting and projecting3--if two groups of three conics each be passed through three nodes and tangent to the quartic1 their six points of intersection 7three of each group7 lie on another connodal trinodal quartic4 ,if the two triangles are inscribed in a conic1 their six sides touch another conic4 ,inverting and projecting3--if two groups of three conics each be passed through the three nodes of a quartic so that the three points of intersection of each group lie on the quartic1 these six conics all touch another connodal trinodal quartic4 ,a triangle is circumscribed about one conic1 and two of its angular points are on a second conic2 the locus of its third angular point is a conic4--,inverting and projecting3--if three conics be drawn through the three nodes of two connodal trinodal quartics so that they all touch one of the quartics and two of their points of intersection are on the other quartic1 the locus of their third point of intersection is a connodal trinodal quartic4 ,a triangle is inscribed in one conic and two of its sides touch a second conic2 the envelope of its third side is a conic4 ,inverting and projecting3--if three conics be drawn through the three nodes of two connodal trinodal quartics so that their three points of intersection lie on one of the quartics and two of them touch the other quartic1 the envelope of the third conic is another connodal trinodal quartic4 ,the theorems of this section are stated in the most general terms and are still true when one or more of the nodes are changed into cusps4 ,it is therefore not necessary to give separate theorems for the case of one cusp and two nodes4 ,,nodal ,,bicuspidal ,,quartics4 ,a quartic with one node and two cusps is a curve of the fourth class1 having one double tangent and two points of inflection 7see ,salmon74 ,hence its reciprocal is also a nodal bicuspidal quartic1 a fact of which frequent note will be made in this section4 ,the inverse of a conic with respect to a focus is a curve called ,pascal's ,limaon4 ,from the polar equation of a conic1 the focus being the pole1 it is evident that the polar equation of the limaon may be written in the form3 e #a r ---- cosx ---- 2 p p where e and p are constants1 being respectively the eccentricity and semi-latus rectum of the conic4 ,from the above equation it is readily seen that the curve may be traced by drawing from a fixed point ,o on a circle any number of chords and laying off a constant length on each of these lines1 measured from the circumference of the circle4 ,the point ,o is the node of the limaon2 and the fixed circle1 which ,i shall call the base circle1 is the inverse of the directrix of the conic4 ,this is readily shown as follows3--the polar equation of the directrix is r p_/7e cosx74 ,hence the equation of its inverse is r 7e cosx7_/p1 which is the equation of the base circle of the limaon4 ,the envelope of circles on the focal radii of a conic as diameters is the auxiliary circle4 ,inverting3--the envelope of perpendiculars at the extremities of the nodal radii of a limaon is a circle with its centre on the axis and having double contact with the limaon4 ,projecting3--from any point on a nodal bicuspidal quartic draw lines to the three nodes and a fourth line forming with them a harmonic pencil2 the envelope of all such lines is a conic through the two cusps and having double contact with the quartic2 the chord of contact passes through the node and cuts the line joining the cusps so that this point of intersection1 the two cusps1 and intersection of the double tangent with the cuspidal line form a harmonic range4 ,reciprocating3--on any tangent to a nodal bicircular quartic take the three points where it cuts the two inflectional tangents and the double tangent1 and a fourth point forming with these a harmonic range2 the locus of all such points is a conic touching the two inflectional tangents and having double contact with the quartic2 the pole of the chord of contact is on the double tangent2 join this last point to the intersection of the inflectional tangents and join the node with the same intersection2 these four lines form a harmonious pencil4 ,if the tangent at any point ,p of a conic meet the directrix in ,q1 the line ,p ,q will subtend a right angle at the focus ,o2 the circle ,p ,o ,q has ,p ,q for a diameter and hence cuts the conic at ,p at right angles4 ,inverting3--from any point ,p on the limaon draw ,o ,p to the node ,o2 draw ,o ,q perpendicular to ,o ,p meeting the base circle in ,q2 ,p ,q is normal to the limaon at ,p4 ,projecting3--from any point ,p on a nodal bicuspidal quartic draw lines to the three nodes and a fourth harmonic to these three2 from ,o draw lines to the two cusps and a fourth harmonic to these two and the line ,o ,p2 the locus of the intersection of the fourth line of each pencil is a conic through the three nodes4 ,call this the basal conic of the quartic4 ,reciprocating3--on any given tangent to a nodal bicuspidal quartic take its points of intersection with the double tangent and the inflectional tangents1 and a fourth point harmonic with these2 on the double tangent take its points of intersection with the given tangent and the inflectional tangents1 and a fourth point harmonic with these2 the envelope of the lines joining the fourth point of these two ranges is a conic touching the double and inflectional tangents4 ,the locus of the foot of the perpendicular from the focus on the tangent to a conic is the auxiliary circle4 ,inverting3--draw a circle through the node tangent to a limaon2 draw the diameter ,o ,p of this circle2 the locus of ,p is a circle having double contact with the limaon1 the axis being the chord of contact4 ,cor42 the locus of the centre of the tangent circle is also a circle4 ,projecting3--through the three nodes of a nodal bicuspidal quartic draw any conic touching the quartic2 the locus of the pole with respect to this conic of the line joining the two cusps is a conic2 draw the chord ,o ,p of the first conic through the node ,o and the pole of the line joining the two cusps2 the locus of ,p is a conic through the cusps1 having double contact with the quartic4 ,if chords of a conic subtend a constant angle at the focus1 the tangents at the ends of the chords will meet on a fixed conic1 and the chords will envelope another fixed conic2 both these conics will have the same focus and directrix as the given conic4 ,inverting3--draw two nodal radii of a limaon ,o ,p and ,o ,q1 making a given angle at ,o2 the envelope of the circle ,p ,o ,q is another limaon2 the locus of the intersection of circles through ,o tangent to the limaon at ,p and ,q is another limaon4 ,these two limaons have the same node and base circle as the given one4 ,projecting3--through the node ,o of a nodal bicuspidal quartic draw a pencil of radii in involution2 let ,o ,p and ,o ,q be a conjugate pair of these nodal radii2 the envelope of the conic through ,p1 ,q1 and the three nodes1 is another quartic of the same kind3 also draw conics through the three nodes tangent to the quartic at ,p and ,q2 the locus of their point of intersection is another quartic of the same kind4 ,these three quartics all have the same node1 cusps1 and base conic4 ,every focal chord of a conic is cut harmonically by the curve1 the focus1 and directrix4 ,inverting3--every nodal chord of a limaon is bisected by the base circle4 ,projecting3--every nodal chord of a nodal bicuspidal quartic is cut harmonically by the quartic1 the base conic1 and the line joining the two cusps4 ,reciprocating3--from any point on the double tangent of a nodal bicuspidal quartic draw the other two tangents to the quartic and a line to the intersection of the inflectional tangents2 the fourth harmonic to these lines envelopes a conic4 ,since the limaon is symmetrical with respect to the axis1 it follows that the two points of inflection are situated symmetrically with respect to the axis4 ,hence the line joining the two points of inflection is parallel to the double tangent4 ,therefore by projection we infer the following general theorem for the nodal bicuspidal quartic3 the line joining the two cusps1 the line joining the two points of inflection1 and the double tangent meet in a point4 ,also the fourth harmonic points on each of these lines lie on a line through the node4 ,reciprocating3--the point of intersection of the cuspidal tangents1 the point of intersection of inflectional tangents1 and the node all lie on a right line4 ,from the node draw a fourth harmonic to this right line and the tangents at the node2 draw a fourth line harmonic to this right line and the inflectional tangents2 draw a fourth harmonic to the cuspidal tangents and this right line2 these three lines all meet in a point on the double tangent4 ,,tricuspidal ,,quartics4 ,a tricuspidal quartic is a curve of the third class with one double tangent and no inflection4 ,its reciprocal is therefore a nodal cubic4 ,we shall begin by reciprocating some of the simpler properties of nodal cubics4 ,since the three points of inflection of a nodal cubic lie on a right line1 it follows that the three cuspidal tangents of a tricuspidal quartic meet in a point4 ,the reciprocal of the harmonic polar of a point of inflection is a point on the double tangent1 found by drawing through the point of intersection of the three cuspidal tangents a line forming with them a harmonic pencil4 ,three such lines can be drawn and it is not difficult to distinguish them4 ,all six lines form a pencil in involution1 the lines to the points of contact of the double tangent being the foci4 ,i shall call such a point on the double tangent the harmonic point of the cuspidal tangent4 ,since any two inflectional tangents of a nodal cubic meet on the harmonic polar of the third point of inflection1 it follows that any two cusps of a trinodal quartic and the harmonic point of the third cuspidal tangent lie on a right line4 ,since the point of contact of the tangents from a point of inflection of a nodal cubic is on the harmonic polar of the point1 it follows that the tangent to the tricuspidal quartic at the point where it is cut by a cuspidal tangent passes through the harmonic point of that cuspidal tangent4 ,the inverse of the parabola from a focus is the cardioid2 and the inverse of the corresponding directrix is the base circle of the cardioid4 ,the cardioid projects into a tricuspidal quartic and its base circle projects into a conic through the three cusps which has the same general properties as the base conic of the nodal bicuspidal quartic4 ,the circle circumscribing the triangle formed by the three tangents to a parabola passes through the focus4 ,inverting3--three circles through the cusp1 and tangent to a cardioid1 intersect in three collinear points4 ,projecting3--three conics through the three cusps of a tricuspidal quartic and touching the quartic intersect in three collinear points4 ,reciprocating3--if three conics touch the three inflectional tangents of a nodal cubic and the cubic itself1 their three other common tangents intersect in a point4 ,circles described on the focal radii of a parabola as diameters touch the tangent through the vertex4 ,inverting and projecting3--from a point on a tricuspidal quartic lines are drawn to the three cusps and a fourth line forming a harmonic pencil2 the envelope of this fourth line is a conic through the three cusps and touching the quartic at the point where the latter is cut by one of the cuspidal tangents4 ,there are three such conics1 one corresponding to each cusp4 ,at any cusp the tangent to its corresponding base conic1 the cuspidal tangent1 and the lines to the other two cusps form a harmonic pencil4 ,reciprocating3--on any tangent to a nodal cubic take the three points of intersection with the inflectional tangents and a fourth point forming with these a harmonic range2 the locus of this fourth point is a conic touching the three inflectional tangents and the cubic4 ,the tangent to the cubic where it is touched by the conic goes through a point of inflection4 ,on any inflectional tangent the point of contact of this conic1 the point of inflection1 and the points of intersection of the other two inflectional tangents form a harmonic range4 ,the circle described on any focal chord of a parabola as diameter will touch the directrix4 ,inverting3--the circle described on any cuspidal chord of a cardioid will touch the base circle4 ,projecting3--through a cusp ,c draw any chord of a tricuspidal quartic meeting the quartic in ,p and ,o2 draw a conic through ,p1 ,o1 and the other two cusps so that the pencil at ,p formed by the tangent to the conic and the lines to the cusps is harmonic2 all such conics will touch the base conic of the cusp ,c4 ,reciprocating3--from ,o1 on any inflectional tangent of a nodal cubic1 draw two tangents ,p and ,q to the cubic2 draw a conic touching the tangents ,p and ,q and the other two inflectional tangents so that the range on one of these tangents formed by the point of contact of the conic and the intersection of the three inflectional tangent is harmonic2 the envelope of all such conics is a conic touching the three inflectional tangents4 ,the directrix of a parabola is the locus of the intersection of tangents at right angles to one another4 ,inverting and projecting3--through any point ,p on the base conic of a cusp ,c of the tricuspidal quartic1 two conics can be drawn through the three cusps and touching the quartic2 their two tangents at ,p and the lines to the other two cusps form a harmonic pencil2 their two points of contact lie on a line through ,c4 ,reciprocating3--from any point on one of the inflectional tangents to a nodal cubic draw the two tangents ,p and ,q2 draw two conics each touching the cubic and the three inflectional tangents1 one touching ,p and the other ,q2 the envelope of their other common tangent is a conic touching the three inflectional tangents2 the two points of contact of any one of these common tangents and the points where it cuts the other two inflectional tangents form a harmonic range4 ,any two parabolas which have a common focus and their axes in opposite directions cut at right angles4 ,inverting3--any two cardioids having a common cusp and their axes in opposite directions cut at right angles4 ,projecting3--two tricuspidal quartics having common cusps and at one of the cusps the same cuspidal tangent1 but the cusps pointed in opposite directions1 cut at such an angle that the tangents at a point of intersection and the lines to the other two cusps form a harmonic pencil4 ,reciprocating3--two nodal cubics have common inflectional tangents and on one of them the points of inflection common1 but the branches of the curve on opposite sides of the line2 any common tangent to the two curves is cut harmonically by the points of contact and the other two inflectional tangents4 ,circles are described on any two focal chords of a parabola as diameters2 their common chord goes through the vertex of the parabola4 ,inverting3--circles are described on any two cuspidal chords of a cardioid2 the circle through their points of intersection and the cusp goes also through the vertex of the cardioid4 ,projecting3--through one of the cusps of a tricuspidal quartic draw two chords2 draw conics through the other two cusps and the extremities of each of these chords so that the pole of the line joining the other two cusps with respect to each of these conics is on the corresponding chord2 the conic through the points of intersection of these two conics and the cusps passes also through the point where the cuspidal tangent of the first mentioned cusp cuts the quartic4 ,reciprocating3--on one of the inflectional tangents1 of a nodal cubic take two points ,p and ,q2 draw a pair of tangents from each of these points to the cubic2 draw two conics each touching a pair of these tangents and the other two inflectional tangents1 so that the polars of the point of intersection of the other two inflectional tangents with respect to each of those conics pass respectively through ,p and ,q2 the conic touching the common tangents to these two conics and the three inflectional tangents touches also the tangent from the first mentioned point of inflection to the cubic4 ,,quartics ,,with ,a ,,triple ,,point4 ,since a triple point is analytically equivalent to three double points1 a quartic with a triple point is unicursal4 ,such a quartic is obtained by inverting a unicursal cubic from its node4 ,the equation of such a cubic may be written u u #j1 where u and u are homogeneous functions of the second and third degree respectively in x and y4 ,hence the equation of the inverse curve is u u 7x y71 which shows that the origin is a triple point and the quartic circular4 ,by projecting this all other forms may be obtained4 ,the nature of the triple point depends upon the relation of the line at infinity to the cubic before inversion4 ,thus the line at infinity may cut the cubic in three distinct points all real1 or one real and two imaginary1 in one real and two coincident points 7an ordinary tangent71 or in three coincident points 7an inflectional tangent74 ,hence the quartic may have at the triple point three distinct tangents all real1 or one real and two imaginary1 one real and two coincident1 or all coincident4 ,this quartic may be generated in a manner similar to that used for the curves already discussed4 ,we showed in the section on nodal cubics that a system of conics through ,a1 ,b1 ,c1 ,d1 and a projective pencil of rays with its vertex at ,a generate by the intersection of corresponding elements a cubic with a node at ,a4 ,invert the whole figure from ,a and then project3--the pencil of rays remains a pencil2 the system of conics becomes a system of unicursal cubics having a common node at ,a and passing through five other common points2 the cubic inverts and projects into a quartic with a triple point at ,a1 passing through the five other common points of the system of cubics4 ,the three points of inflection of a nodal cubic lie on a right line4 ,inverting3--there are three points on a circular quartic with a triple point whose osculating circles pass through the triple point1 and these three points lie on a circle through the triple point4 ,let these three points be designated by ,a1 ,b1 and ,c4 ,the lines from the triple point ,o to the points ,a1 ,b1 ,c1 and the common chord of the osculating circles at two of them form a harmonic pencil4 ,through one of these points1 ,a1 and the triple point draw a circle touching the quartic2 the point of contact is on the common chord of the osculating circles at ,b and ,c4 ,from theorems which we have already proved for a system of cubics having a common node and passing through five others fixed points1 we can infer other theorems for a system of quartics having a common triple point and passing through seven other fixed points4 ,for example1 any conic through the common double point and two of the fixed points is cut by the cubics in pairs of points which determine at the node a pencil in involution4 ,hence any cubic having its node at the common triple point and passing through any four of the fixed points is cut by the quartics in pairs of points which determine at the common triple point a pencil in involution4 ,again1 the pairs of tangents to the cubics at the common double point form a pencil in involution1 the two cuspidal tangents being the foci of the pencil4 ,inverting3--the line at infinity 7which passes through two of the fixed points1 i4 e4 the circular points7 cuts the system of circular quartics in pairs of points in involution4 ,projecting3--a line through any two of the seven fixed points cuts the system of quartics in pairs of points in involution4 ,since the line at infinity touches the inverse of a cuspidal cubic1 it follows that any line through two of the fixed points will touch two of the quartics of the system2 these points of contact are therefore the foci of the involution4 ,other theorems on such a system of quartics will be given in the next section4 ,,systems ,,of ,,quartics ,,through ,,sixteen ,,points4 ,let ,u and ,v represent a system of quartics through sixteen points4 ,since the discriminant of quartic is of the #bgth degree in the coefficients it follows that there are #bg values of k for which the discriminant vanishes1 and hence #bg quartics of the system which have double points4 ,as in case of cubics these #bg points are called the critic centres of the system4 ,let the equation of the system of quartics be written u u u u u #j4 ,in a manner similar to that employed for cubics1 we find the equation of the polar cubics of the origin with respect to the system to be u #bu #cu #du #j4 ,the polar conics of the origin are given by u #cu #fu #j2 and the polar lines of the origin1 by u #du #j4 ,the origin may be any point in the plane and hence we conclude that only one quartic of the system passes through a given point and that the polar cubics of any point form a system through nine points4 ,the polar conics of any point form a system through four points and the polar lines meet in a point4 ,if one of the critic centres be taken for origin1 we can readily see that such a point is also a critic centre on each of its systems of polar curves4 ,it is thus at a vertex of the self-polar triangle of its system of polar conics and the opposite side of the triangle is the common polar line of the critic centre with respect to each of the systems of curves4 ,the tangents at the node of the nodal quartic coincide with those of its polar cubic and these we know coincide with the lines which constitute its polar conic4 ,if two of the sixteen basal points coincide1 such a point is a critic centre4 ,the argument is the same as for a system of cubics4 ,we can also see that two of the basal points of each of its systems of polar curves coincide at the critic centre4 ,the sixteen basal points of the system of quartics may unite two and two so that it is possible to draw a system of quartics touching eight given lines each at a fixed point4 ,if three of the basal points of our system of quartics coincide1 all the quartics have at such a point a common point of inflection and a common inflectional tangent4 ,the demonstration is the same as that already given for cubics4 ,the system of polar cubics of such a point also have this point for a common point if inflection and the same tangent for a common inflectional tangent4 ,i prefer to show this analytically for the sake of the method4 ,the equation of the system of quartics having the origin for a common point of inflection and the axis of y for a common inflectional tangent may be written u u 7,b k,b7xy 7,c k,c7y 7,a k,a7y #j4 ,the equation of the polar cubics of the origin is therefore1 u #b7;,b k,b7xy 7,c k,c7y #c7;,a k,a7y #j1 which proves the proposition4 ,the properties of the system of polar conics of such a point are therefore the same as those already proved for cubics4 ,one quartic of the system has a double point at the common point of inflection of the others4 ,when four basal points coincide they give rise either to a common point of undulation or a common double point on all the quartics of the system4 ,the equation of the system having a common point of undulation may be written u 7,a k,a7xy 7,b k,b7xy 7,c k,c7y 7,d k,d7xy 7,e k,e7y 7,f k,f7y #j4 ,there is one value of k for which the last term vanishes1 and hence the origin is a critic centre4 ,the polar cubics of the point of undulation break up into a system of conics through four points and the common tangent at the common point of undulation4 ,for the equation of the polar cubics is y7,a k,a7x 7,b k,k7xy 7,c k,c7y #b7;,d k,d7x #b7;,e k,e7y 7,f k,f7 #j4 ,the system of polar conics of the origin consequently breaks up into the line y #j and a pencil meeting in a point4 ,the common tangent at the common point of undulation is also the common polar line of the point of undulation4 ,when the four coincident basal points form a common double point on the quartic1 it is not difficult to show that two of the quartics are cuspidal at this point4 ,the polar cubics of the common double point form a system having the same point for common double point4 ,the tangents to the quartics at the common node constitute the system of polar conics and form a pencil in involution4 ,twelve of the sixteen basal points may unite in three groups of four each and the system of quartics is then trinodal and passes through four other fixed points4 ,this is the system obtained by inverting a system of conics through four points and then projecting4 ,a few special cases should be noticed here4 ,if the four fixed points and two of the nodes lie on a conic1 this conic together with the two lines from the third node to the first two constitute a quartic of the system4 ,if the four fixed points lie on a line1 the quartic then consists of this line and the sides of the triangle formed by the nodes4 ,if the three nodes and three of the fixed points lie on a conic1 the system of quartics then consists of this conic and a system of conics through the three nodes and the fourth fixed point4 ,a special case of a system of quartics with three nodes is a system of cubics having a common node and passing through five other fixed points together with a line through two of them4 ,if a fifth basal point be moved up to join the four at the common node1 the quartics have one tangent at the common node common to all4 ,if six basal points coincide they have both tangents at the node common to all4 ,in this case one of the quartics has a triple point at the common node of the others4 ,if seven basal points coincide1 one of these tangents is an inflectional tangent as well4 ,if eight points coincide1 both are inflectional tangents4 ,when nine of the basal points of a system of quartics coincide1 the quartics have a common triple point4 ,this is nicely shown by inverting a system of nodal cubics from the common node4 ,the inverse curves form a system of quartics having a triple point and passing through seven other fixed points4 ,the common triple point on two quartics counts for nine points of intersection and the seven others make the requisite sixteen4 ,from our knowledge of a system of cubics having a common node it is readily inferred that three of the quartics must each break up into a nodal cubic and a right line through the node4 ,if the seven fixed points of the system of quartics lie on a cubic having a node at the common triple point1 the system of quartics then consists of this cubic and a pencil of lines through the node4 ,if two of the seven fixed points lie on a line through the common triple point1 the system of quartics then consists of this right line and a system of cubics through the other five points and having a common node at the common triple point4 ,the system of cubics having a common node may have one1 two1 or three of the other basal points at infinity2 and these may be all distinct or two or three of them coincident4 ,whence we infer that if the system of quartics have ten coincident basal points1 one of the tangents at the triple point is common to all the quartics of the system4 ,if eleven basal points coincide1 two of the triple-point tangents are common to all the quartics4 ,if twelve coincide1 all three triple-point tangents are common4 ,these triple-point tangents may be all distinct1 two coincident1 or all three coincident4 ,if thirteen basal points coincide1 the system of quartics then consists of the three fixed lines joining the multiple point to the other three1 together with a pencil of lines through the multiple point4 ,if fourteen points coincide1 two lines are fixed and these with any two lines of the pencil form a quartic of the system4 ,if fifteen points coincide1 only one line is fixed and each quartic consists of this line and any other three of the pencil4 ,when all sixteen points coincide1 any four lines through it form a quartic of the system4 ,in this paper cubic and quartic curves only are considered4 ,i expect in a future paper to extend the methods herein developed to curves of still higher degrees4 ,many of the present results can be generalized and stated for a unicursal curve of the nth degree4 ,i have purposely omitted all consideration of focal properties of these curves4 ,there are also many special forms of interest which do not properly belong to a general treatment of the subject4 ,,note ,a4 ,the theorem concerning the three points on a conic ,a1 ,b1 and ,c1 whose osculating circles pass through a fourth point ,o on the conic1 is due to ,steiner4 ,from the properties of the harmonic polars of the points of inflection on a nodal cubic we may infer many other theorems concerning the points ,a1 ,b1 and ,c on a conic4 ,let the cubic be projected into a circular cubic and then inverted from the node4 ,its points of inflection ,a1 ,b1 ,c invert into the points ,a1 ,b1 and ,c4 ,the harmonic polar of ,a inverts into the common chord ,o ,p of the circles osculating the conic at ,b and ,c2 and similarly for the other harmonic polars4 ,the pencil ,o ,a ,b ,p ,c is harmonic4 ,any circle through ,a and ,o meets the conic in ,s and ,t so that the pencil ,o ,a ,s ,p ,t is harmonic4 ,the two circles through ,o and tangent to the conic at ,s and ,t intersect on ,o ,p4 ,if two circles be drawn through ,o and ,a intersecting the conic one in ,s and ,t and the other in ,u and ,v1 the circles ,o ,s ,u and ,o ,t ,v intersect on ,o ,p2 so also the circles ,o ,s ,v and ,o ,t ,u4 ,but one circle can be drawn through ,o and ,a and tangent to the conic2 its point of contact is on ,o ,p4 ,let l1 m1 and n be three points on the conic on a circle through ,o4 ,draw the circles ,o ,a l1 ,o ,a m1 and ,o ,a n intersecting the conic again in l1 m1 n2 l1 m1 n1 are also on a circle through ,o1 and the circles through l1 m1 n and l1 m1 n intersect on ,o ,p4 ,,note ,b4 ,from the fundamental property of the ,cissoid of ,diocles we can obtain by inversion an interesting theorem concerning the parabola4 ,in the figure of the ,cissoid given in ,salmon's ,h4 ,p4 ,c4 ,art4 #bad1 ,a ,m ,m ,r1 whence ,a ,m ,a ,r - ,a ,m2 or ,a ,r ,a ,m ,a ,m4 ,inverting from the cusp and representing the inverse points by the same letters1 we have for the parabola #a #a #a ------ ------ ---------- 4 ,a ,r ,a ,m ,a ,m ,this result is interpreted as follows3--draw the circle of curvature at the vertex of a parabola2 this circle is tangent to the ordinate ,b ,d which is equal to the abscissa ,a ,d2 draw a line through ,a cutting the circle in ,r1 the ordinate ,b ,d in ,m1 and the parabola in ,m2 then #a #a #a ------ ------ ---------- 4 ,a ,r ,a ,m ,a ,m ,draw the circle with centre at ,d and radius ,a ,d2 any chord of the parabola through the vertex is cut harmonically by the parabola1 the circle1 and the double ordinate through ,d4 ,footnotes3 #a ,see note ,a4 #b ,a few of the results of this section are due to the late ,mr4 ,h4 ,b4 ,hall4 #c ,see note ,b4 ,foreign ,settlements in ,kansas4 ,a ,,contribution ,,to ,,dialect ,,study ,,in ,,the ,,state4 ,explanatory4--,some years ago when the subject of dialect study in ,kansas1 or rather of ,kansas dialect1 was mentioned1 ,mr4 ,noble ,prentis1 a gentleman who is warranted in speaking with authority on ,kansas1 was inclined to think that he settled the question in short order by declaring that there is no ,kansas dialect4 ,probably the majority of intelligent citizens of the state would turn off the subject with the same reply4 ,in the sense of a mode of speech common to the inhabitants of ,kansas and peculiar to them1 ,mr4 ,prentis was indeed right4 ,there is no vocabulary1 at least no extensive vocabulary1 by which the native of ,kansas may be recognized in the ,american ,babel4 ,we have no distinctive pronunciation by which we may be known from the inhabitants of ,nebraska or set apart from the citizens of ,missouri4 ,the verb fails to agree with its subject and the participle is deprived of its final #8g' with about equal frequency in ,western ,kansas and ,eastern ,colorado4 ,but in the same sense it is true that there is no ,kansas flora1 no ,kansas fauna2 that is1 there is no plant and there is no animal found quite generally in ,kansas and found nowhere outside of ,kansas4 ,the remark that there is no such thing as a ,kansas dialect rests upon a misapprehension of what is meant by the term4 ,in just the same way that we speak of the flora and the fauna of ,kansas we may speak of the dialect of ,kansas4 ,yet to avoid popular misapprehension it may be better to speak of dialect in ,kansas1 rather than of ,kansas dialect4 ,dialect study involves the observation and description of all facts concerning the natural living speech of men1 and especially those points in which the speech of individuals or groups differs from that of the standard literary language as represented in classic writers and classic speakers4 ,standard literary ,english is always a little behind the times4 ,it is the stuffed and mounted specimen in the museum4 ,dialect is the live animal on its native heath4 ,most people1 indeed1 will think that their speech does not differ materially from standard ,english4 ,they say1 8,we speak near enough alike #8for practical purposes'4 ,but a thousand years hence the pronunciation of our country may have changed so much that it will seem like another language1 and our descendants will write learned theses to prove that we pronounced #8cough' like cow or like cuff4 ,a new language will have grown out of an old one1 but no one know how it came about4 ,careful dialect study will help explain it40 ,kansas is a peculiarly favorable field for dialect study4 ,we have here side by side representatives from nearly every state in the ,union1 and from a dozen foreign countries4 ,the observer has here what elsewhere he must travel over half the world to find4 ,in a district where the people are all natives1 the speech is so nearly homogeneous that it is difficult to find any one who recognizes the peculiarities of his own language1 but here the contrast of strange tongues strikes us immediately and we become conscious early of the fact that all men do not speak alike4 ,study of dialect may be classified under the heads of pronunciation1 grammar and vocabulary4 ,of these the last two are the easiest1 and may be carried on by almost any one with pleasure and valuable results4 ,pronunciation is the most difficult of these matters to study1 as competent observation and reports can be made only by one who has made a thorough study of ,phonetics4 ,to those who might wish to take up the study of this branch of the subject1 ,sweet's ,primer of ,phonetics1 and ,grandgent's 8,vowel ,measurements0 and 8,german and ,english ,sounds0 are recommended4 ,in the study of dialect vocabularies it may become of the greatest importance to establish the exact locality of a word and the origin of the persons by whom it is used4 ,for instance1 in a family of my acquaintance the word #8slandering' sauntering was familiar4 ,it was a great puzzle to me until ,i learned that some of the children had been in the care of a ,german maid4 ,the ,german word #8schlendern' suggested the unquestionable source of the peculiar word4 ,as a source of information regarding the origin of the foreign elements of our population when their native speech shall have been forgotten1 but when the influence of it will be left in vocabulary and pronunciation ,i have thought that a map of the state with the location of all the foreign settlements of even quite small size would be of interest and in time of great value4 ,in the following pages ,i transmit the results of my inquiries so far as received4 ,it is my intention to make the report complete and to publish the map1 when as complete as it can be made1 in colors4 ,unexpected difficulties have delayed the work and prevented its being complete4 ,i depended for my information upon the ,county ,superintendents of the ,state1 a class of unusually intelligent and well-informed men and women4 ,but in not a few cases there seems to have been a suspicion in the mind of my correspondent that ,i might be a special officer of the state trying to locate violations of the law requiring district schools to be conducted in ,english1 and hence information regarding schools in foreign tongue was withheld or given but partially4 ,and in some cases my informants were not well posted4 ,a superintendent by the name of ,schauermann in a county containing a town called ,suabia1 tells me that there are no foreigners in his county4 ,in such cases time must be taken to secure a correct result4 ,the questions asked were3 ,locate1 and give origin1 date and approximate numbers of any settlements--six or more families--of foreigners in your county4 ,do they still use their language to any extent8 ,do they have church service and schools conducted in their native tongue8 ,in many replies one or more of these points was neglected so that the information is not yet by any means what ,i desire to make it4 ,however1 for the purpose of dialect study approximate correctness in location is of chief importance1 and accuracy as to numbers quite secondary4 ,through the aid of ministers and others to whom ,i have been referred by the superintendents ,i hope to make this report complete in the following respects3 ,the more exact limits of the settlement2 the numbers of those foreign-born2 the province as well as land from which they came2 the number of churches2 the number of schools and the length of time the same are conducted4 ,i solicit the co-operation of everyone interested in this work1 and also in the whole subject of dialect study4 ,as intimated above1 interested observers can without especial training do a service to science and at the same time find a fascinating pastime for themselves by making collections of words and constructions which they believe to be unusual or new4 ,if any such are sent to the writer they will be duly acknowledged4 ,they should in every case be accompanied by a statement of the age1 condition and birth-place of the person using them4 ,i wish here to call attention to the work of the ,american ,dialect ,society which exists to promote this study4 ,it desires as wide a membership as possible1 and membership is open to all interested in the subject4 ,the publication of the ,society1 ,dialect ,notes1 contains reports of word-lists and other studies1 and will be an aid to any who wish to undertake similar work4 ,subscriptions and membership fees should be sent to ,mr4 ,c4 ,h4 ,grandgent1 ,treas41 ,cambridge1 ,mass4 ,,reports ,,by ,,counties4 ,,atchison4--,reports no foreigners1 by ,john ,klopfenstein1 ,supt4 ,,allen4--,swedes and ,danes1 from #fjj to #gjj1 settled from #ahgc to #ahhj4 ,have church service1 and four to five months school in ,swedish4 ,grove and ,elsmore townships4 ,germans in and around ,humboldt4 ,,anderson4--,irish in ,reeder township1 #ahfj and #ahgd4 ,germans1 #ahfj in ,putman township1 #ahhj in ,westphalia township4 ,have both church and schools in ,german4 ,,barber4--,reports 8no foreigners worth making account of01 by ,j4 ,o4 ,hahn1 ,sup't4 ,,barton4--,no report4 ,,bourbon4--,reports no foreigners4 ,,brown4--,no report4 ,,butler4--,germans 7,prussians71 speaking ,low ,german1 in ,fairmount and ,milton townships4 ,hold church services but no schools in ,german4 ,,chase4--,russian ,mennonites1 speaking both ,russian and ,german1 in ,diamond ,creek township1 no church1 but a portion of schooling in ,german4 ,germans at ,strong ,city1 with both church and schools in their native tongue4 ,,chatauqua4--,some ,norwegians and ,swedes1 #ahgj1 no location given4 ,neither schools nor churches in native tongue4 ,one colony of #8,russians' 7,mennonites871 who have also given up their language4 ,,cherokee4--,weir ,city1 ,french and ,italians1 number considerable4 ,scammon1 ,scotch1 also in large numbers4 ,the ,french and ,italians have neither schools nor church in the native tongue4 ,germans in ,ross1 twenty families2 with church originally ,lutheran1 now ,mennonite2 school irregularly during past ten years4 ,swedes1 a few families in ,cherokee township1 have entirely given up ,swedish language4 ,the ,scotch1 ,french and ,italians in mines or mining industries4 ,,cheyenne4--,germans settled in #ahhe-hf on ,hackberry ,creek1 #afj persons2 in the northeast corner of the county1 #ajj2 on west border of county1 north of ,republican river1 #abj2 all with churches and the last two with occasional schools4 ,swedes are across the ,republican adjoining last named ,german settlement1 #abj1 entered #ahhf1 having neither church nor school in ,swedish4 ,,clarke4--,reports no foreigners4 ,,clay4--,no report4 ,,cloud4--,canadian ,french are scattered over much of the county1 with considerable settlements in and around the towns of ,concordia1 ,clyde1 ,st4 ,joseph and ,aurora4 ,in all there are churches1 in the first three schools also conducted in ,french4 ,norwegians occupy portions of ,sibley and ,lincoln townships with two churches in their own tongue4 ,they number about three hundred4 ,irish occupy portions of ,solomon and ,lyon1 the south part of ,meredith and the southeast corner of ,grant townships4 ,,coffey4--,germans in ,liberty and on border of ,leroy and ,avon township4 ,have church service in ,german4 ,,comanche4--,germans4 ,a few scattered families4 ,,cowley4--,a few ,swedes and ,germans1 widely scattered4 ,,crawford4--,irish and ,french in ,grant township2 ,swedes in west part of ,sherman township1 have all given up their language4 ,italians1 ,austrians and other nationalities in south part of ,washington1 southeast part of ,sheridan and all over ,baker township1 especially in ,pittsburg1 employed in mining and smelting4 ,,decatur4--,swedes in ,oberlin township2 ,mennonites in ,prairie ,dog township2 ,germans in and around ,dresden1 with ,catholic church2 ,bohemians in ,jennings and ,garfield townships4 ,,dickinson4--,germans1 #ejj in number settled in #ahfj in ,liberty1 ,union and ,lyon townships4 ,have three churches and two schools in ,german4 ,also in ,wheatland1 ,jefferson1 ,bonner and ,ridge townships1 one church and a school4 ,swedes1 #ajj settled in #ahfj in ,center and ,hayes township1 with two churches and one school in ,swedish4 ,irish1 several hundred in south part of ,banner township4 ,,doniphan4--,germans in ,wayne1 ,marion and southern part of ,center1 ,burr ,oak and ,washington townships1 with church service in native tongue4 ,norwegians in eastern part of ,wolf ,river township4 ,,douglas4--,there are ,german settlements in ,eudora township 7#cjj71 ,marion township 7#fjj71 and ,big ,springs township 7#ajj71 with churches in all and school in the first4 ,there are about five hundred ,germans in ,lawrence1 with three ,german churches4 ,there are smaller settlements of ,germans and ,scandinavians at several points in the county4 ,,edwards4--,germans and ,swedes in ,kinsley1 ,jackson and ,trenton townships1 have church service in their mother tongue4 ,,elk4--,swedes in ,painter and ,hood townships2 ,irish in ,falls township2 ,germans on the border of ,elk and ,wild ,cat townships4 ,none of these have church or school in the native tongue1 but all use it at home4 ,,ellis4--,germans from ,russia1 settled about #ahgf in ,catherine1 ,hartsook1 ,lookout1 ,wheatland and ,freedom townships1 about #cjjj in number--one third of the population of the county in #ahia4 ,they are ,catholics1 and have both churches and parochial schools conducted in ,german4 ,they are large wheat-growers4 ,,ellsworth4--,germans1 7,methodists7 in south part of ,valley township2 ,germans 7,lutherans7 in north part of ,columbia and ,ellsworth townships2 ,germans 7,baptists7 from ,prussia1 in ,green ,garden and south west corner of ,empire townships4 ,these all have church service1 and the ,lutherans schools in their own tongue4 ,bohemians in ,valley and ,noble townships4 ,,finney4--,reports no foreigners4 ,,ford4--,germans in ,wheatland and ,speareville townships4 ,have church1 and one school conducted in ,german4 ,,franklin4--,no report4 ,,garfield4--,a few scattered families of ,germans4 ,,geary4--,irish 7,connaught7 came into ,jackson1 ,jefferson and ,liberty townships #ahee1 about #aejj in number4 ,germans 7,anhalt7 about #aejj came into ,jefferson1 ,milford and ,lyon townships in #ahfb4 ,about #cjj ,english from ,sussex settled in ,lyon township in #ahgj4 ,the ,germans maintain both churches and schools in ,german4 ,,gove4--,swedes in ,lewis and south part of ,grinnell and south west corner of ,gove townships4 ,,graham4--,a settlement of ,canadian ,french 7#fjj7 was made in adjacent parts of ,wild ,horse and ,morelan townships about #ahhj4 ,they conduct church service but no schools in ,french4 ,,grant4--,reports no foreigners4 ,,gray4--,reports no foreigners4 ,,greeley4--,swedes in the north west part of the county1 have church service and summer school in ,swedish4 ,,greenwood4--,norwegians1 about #bjj1 in south part of ,salem township1 have church in their own tongue4 ,germans in ,shell ,rock township1 about #cjj1 also have church in their own language4 ,,hamilton4--,reports no foreigners4 ,,harper4--,germans about the town of ,harper4 ,hungarians south of ,bluff ,city4 ,both have church service in ,german4 ,about #ajj ,french in ,odell and ,stohrville townships4 ,,harvey4--,germans 7,russian ,mennonites7 from ,odessa1 a few from ,prussia1 the latter speaking ,low ,german4 ,they settled from #ahgd to #ahgf in ,alta and ,garden townships1 in ,pleasant and the eastern part of ,newton townships1 and about ,halstead4 ,they have church and school in ,german1 but speak ,russian also4 ,french in north part of ,emma township1 engaged in raising silk worms4 ,,haskell4--,reports no foreigners4 ,,hodgman4--,germans1 about #cj families1 settled about #ahhd in south east corner of ,sterling township2 have preaching in ,german4 ,swedes in north west corner of ,marena township4 ,,jackson4--,danes in ,netawaca and ,whiting townships2 ,irish in ,washington township2 neither continue to use their native tongue4 ,,jefferson4--,germans 7,swiss7 in ,delaware1 ,jefferson and ,kentucky townships1 maintaining church but no schools in ,german4 ,,jewell4--,swedes1 widely scattered in ,sinclair1 ,allen1 ,ewing and ,ezbon townships4 ,,johnson4--,no report4 ,,kearney4--,no report4 ,,kingman4--,a small settlement of ,germans in ,peters township1 not using ,german to any extent4 ,a few ,irish in ,union township4 ,,kiowa4--,no report4 ,,labette4--,swedes and ,norwegians settled in ,valley and ,canada townships about #ahfi4 ,still speak their language1 but have neither church nor school in it4 ,,lane4--,reports no foreigners4 ,,leavenworth4--,german1 in #ahgc in ,easton township2 in ,fair township in #ahgf2 about #fjj in each place4 ,they have church service and schools in ,german4 ,,lincoln4--,danes settled in ,grant township in #ahfi and since1 #djj in number4 ,germans settled in ,pleasant township in #ahgb1 with #cjj1 and in ,indiana township in #ahfi and later with about #cge4 ,danes and ,germans have good schools and churches in native tongue4 ,bohemians in ,highland township in #ahgh with thirty families4 ,they speak their native tongue1 but have no schools or churches4 ,,linn4--,reports no foreigners4 ,,logan4--,swedes1 about #bjj1 about ,page ,city1 in north part of county4 ,have church and school both in ,swedish4 ,,lyon4--,welsh1 between #ajjj and #aejj are located in and about ,emporia1 with three churches conducted in ,welsh4 ,there is a settlement of ,scandinavians near ,olpe in ,centre township4 ,,marion4--,germans 7,russian ,mennonites71 settled in ,logan1 ,durham1 ,lehigh1 ,risley1 ,menno1 ,west ,branch and ,liberty townships1 from #ahgj to #ahge1 some #ejjj in number4 ,they speak both ,russian and ,german1 and have church service and schools in the latter tongue4 ,bohemians1 about #ejj in number are settled in ,clark township4 ,they speak ,czech and have church service in that language4 ,french to the number of #bjj settled soon after #ahgj on the border of ,grant and ,doyle townships4 ,they speak ,french still1 but have no schools or church service in the language4 ,,marshall4--,germans 7,pommeranians1 ,hanoverians1 ,frisians7 to the number of #bjjj1 settled in the west part of ,marysville township from before #ahfj to #ahgj4 ,they have both church and school in their own tongue4 ,in the north part of ,murray and the south half of ,vermillion townships are #abjj ,irish1 who use only ,english in church and school4 ,they came before #ahgj4 ,bohemians in small numbers occupy the north part of ,guittard1 the north west corner of ,waterville and the south part of ,blue ,rapids townships2 ,swedes a portion of the south part of ,waterville township4 ,no report as to their language4 ,,meade4--,no report4 ,,miami4--,germans occupy the north part of ,wea and the west part of ,valley townships1 about #bjj in each settlement2 the first has a ,catholic1 the second a ,lutheran church4 ,irish occupy the north part of ,osage township1 also about #bjj in number4 ,,mitchell4--,germans to the number of #abjj occupy ,pittsburg1 ,blue ,hill1 and ,carr ,creek townships4 ,in the first there is a church1 and a well-attended school 7,catholic7 at ,tipton4 ,,montgomery4--,germans to the number of #ajj are settled in and about ,independence4 ,they have church service in ,german 7,lutheran74 ,,morris4--,swedes occupy ,diamond ,valley1 the west part of ,creek1 and the north part of ,parker townships4 ,they have several churches and occasionally a school conducted in ,swedish4 ,,morton4--,reports no foreigners4 ,,mcpherson4--,swedes settled1 about #ahgj1 in ,union1 ,smoky ,hill1 ,harper1 ,new ,gottland1 ,delmore1 and portions of other townships1 in large numbers1 several thousand4 ,they have several churches and excellent schools conducted in ,swedish4 ,germans 7,russian ,mennonites7 occupy ,superior1 ,turkey ,creek1 ,mound1 ,lone ,tree1 ,king ,city1 and portions of ,mc,pherson and other southern townships1 with several churches and schools4 ,the ,mennonites number about #ejjj and settled after #ahgf4 ,,nemaha4--,germans 7,swiss7 occupy ,nemaha and ,washington1 and a portion of ,richmond townships1 with ,german churches and schools4 ,irish are in ,clear ,creek and north east corner of ,neuchatel townships4 ,most of ,neuchatel township is occupied by ,french 7,swiss74 ,,neosho4--,germans have a considerable settlement in the south east corner of ,tioga township1 with church service 7,lutheran7 in ,german2 another in the east part of ,lincoln township1 where the language is spoken1 but without church or school4 ,swedes have settlements in the north west part of ,tioga and the east part of ,big ,creek townships2 church in the first only1 though in both ,swedish is spoken almost exclusively4 ,,ness4--,no report4 ,,norton4--,germans to the number of #ajj settled about #ahhj in ,grant township4 ,they have church service in ,german4 ,,osage4--,swedes1 7#gjj in number17 settled in ,grant township in #ahga1 where they have four churches conducted in ,swedish4 ,welsh settled in #ahfi in ,arvonia township1 and others in the north part of ,superior township1 #gjj in number4 ,they have six churches with services in ,welsh4 ,germans are in the north part of ,scranton and ,ridgway townships1 #bjj in number2 ,french in the central part of ,superior township1 #bjj strong2 ,danes1 #bjj1 in north part of ,melvern and ,olivet townships2 a considerable number of ,scotch and ,irish in and near ,scranton4 ,most of these latter are engaged in coal mining4 ,none of the foreigners have schools--except ,sunday schools--in their native tongue4 ,,osborne4--,germans settled in ,bloom township1 where they have both church and school in their mother tongue4 ,,ottawa4--,bohemians are located about the border of ,sheridan and ,fountain townships2 ,danes in the south part of ,buckeye township2 ,irish1 arrived about #ahhe1 in the south part of ,chapman township4 ,none of these have church or school in a foreign tongue4 ,,pawnee4--,swedes settled about #ahgg in the west part of ,garfield and the north part of ,walnut townships1 about #ejj in all4 ,they speak their native language at home almost exclusively1 and have preaching in it4 ,,phillips4--,germans occupy ,mound and south part of ,dayton townships1 with preaching and parochial school in ,german4 ,dutch occupy east part of ,prairie ,view with adjacent portions of ,long ,island1 ,dayton1 and ,beaver townships1 with preaching in ,dutch4 ,some ,danes and ,swedes in ,crystal township1 and some scattered ,poles4 ,,pottawatomie4--,germans1 about #bejj1 in west half of ,mill ,creek and adjacent portions of ,sherman and ,vienna townships1 also in ,pottawatomie and adjacent portions of ,union1 ,louisville1 and ,st4 ,george townships4 ,there are a few families in ,wamego and ,st4 ,mary's ,mission4 ,they have several schools and churches conducted in ,german4 ,swedes occupy the whole of ,blue ,valley and the west border of ,greene townships1 and have a small settlement in ,st4 ,mary township1 numbering in all #abjj4 ,they have church service and a parochial school conducted in ,swedish4 ,irish1 to the number of #bjjj occupy ,clear ,creek1 ,emmet1 ,st4 ,mary and the border of ,st4 ,clere townships4 ,french 7,canadian71 numbering #bjj1 are found in the north part of ,mill ,creek and in ,union townships1 also a few about ,st4 ,mary's ,mission4 ,,pratt4--,reports no foreigners4 ,,rawlins4--,germans in north east part of county with church and school in ,german4 ,swedes in east part of county1 ,bohemians and ,hungarians in north and north east portion4 ,,reno4--,germans1 about #cjj1 came in #ahhj to north east corner of ,little ,river township1 and about #bjj to south east corner of ,sumner township2 also a settlement in the west part of ,hayes township2 ,dutch1 about #cej1 came #ahgh into ,haven township2 ,russians are settled in ,salt ,creek and ,medford townships4 ,all have church service and schools in their native tongue4 ,there are also a few ,french and ,danes in the county4 ,,republic4--,no report4 ,,rice4--,there is a considerable settlement of ,germans in ,valley township1 also ,pennsylvania ,germans in the west part of ,sterling township1 with ,german churches in both4 ,there are also some ,germans in the town of ,lyons1 with a ,german church4 ,,riley4--,swedes1 about #bejj1 occupy ,jackson1 ,swede ,creek and adjacent portions of ,mayday1 ,center1 ,fancy ,creek and ,sherman townships4 ,they have church services and summer schools in their own tongue4 ,bohemians and ,germans1 about #ejj together1 occupy the north east part of ,swede ,creek township4 ,,rooks4--,germans1 #aj families1 settled #ahhj in north part of ,northhampton township4 ,bohemians1 #aj families1 located in north part of ,logan township in #ahgi4 ,french1 about #cj families1 south west corner of ,logan1 and same number in ,twin ,mound township1 settled in #ahgh1 speak ,french and have preaching in that tongue4 ,the ,germans have church service in ,german4 ,,rush4--,germans 7,russian ,mennonites7 are located as follows3 in ,big ,timber township #ge families1 in ,illinois township #be families1 in ,pioneer township #ej families1 in ,lone ,star township #ej families1 in ,banner township #be families1 in ,garfield township #be families1 in ,belle ,prairie township #cj families4 ,in each township there is one church or more1 but no ,german schools 7874 ,bohemians are found in ,banner and ,garfield townships1 about #be families in each4 ,,russell4--,no report4 ,,saline4--,germans1 7,bavarians and ,swabians7 about #bjj1 in ,gypsum and south part of ,ohio townships2 ,swedes1 #cjjj to #djjj1 in ,washington1 ,smolan1 ,falun1 ,liberty and ,smoky ,view1 and adjacent parts of ,spring ,creek1 ,summit and ,walnut townships1 also in ,salina4 ,the ,swedes came in #ahfh4 ,both ,germans and ,swedes have preaching and the latter have schools in their tongue4 ,,scott4--,no report4 ,,sedgwick4--,germans1 #cjjj to #djjj1 settled from #ahgd-hb in ,sherman1 ,grand ,river1 ,garden ,plain1 ,attica and ,union townships4 ,also about #bjjj ,germans in the city of ,wichita4 ,in both places schools and churches in ,german4 ,russians1 ,italians1 ,french and ,scandinavians are represented1 a few hundred each1 in ,wichita4 ,in the country townships a few ,dutch and ,swedes4 ,,seward4--,reports no foreigners4 ,,shawnee4--,germans 7,moravians7 in ,rossville township1 speak their native tongue almost exclusively1 but have neither school nor preaching in ,german4 ,,sheridan4--,no reports4 ,,sherman4--,germans1 #bj families about the center of the county4 ,swedes1 #aj families in north east corner and #be families in south west corner4 ,both ,germans and ,swedes have schools and preaching in their native tongue4 ,,smith4--,germans in west part of ,swan and ,cedar townships1 and on border of ,harvey and ,banner townships1 in both churches1 and in the first schools1 in ,german4 ,dutch1 in the south half of ,lincoln township1 have church but no schools4 ,,stafford4--,germans in ,hayes and ,cooper townships1 three hundred in number1 with two churches having service in ,german4 ,,stanton4--,a few scattered ,germans4 ,,stevens4--,no report4 ,,sumner4--,no report4 ,,thomas4--,a few foreigners scattered about the country2 all anglicised4 ,,trego4--,no report4 ,,wabaunsee4--,germans and some ,swedes in ,kaw1 ,newbury1 ,mill1 ,farmer1 ,alma and ,washington townships1 with both preaching and schools in the mother tongue4 ,,wallace4--,swedes1 to the number of #cjj1 have settled since #ahhh in the south west corner of the county4 ,they have church and schools in ,swedish4 ,,washington4--,germans in ,franklin1 ,charleston1 ,hanover and north part of ,sherman townships1 have both church and schools 7#f7 conducted in ,german4 ,bohemians are numerous in ,little ,blue township2 ,french about midway in ,sherman township2 ,irish in ,barnes1 south part of ,sherman and ,koloko townships4 ,,wichita4--,no report4 ,,wilson4--,swedes have settled since #ahgj in ,colfax township4 ,they have preaching but no schools in ,swedish4 ,,woodson4--,no report4 ,,wyandotte4--,germans1 #aej1 in north west corner of ,prairie township2 ,swedes1 #cej1 in ,kansas ,city1 ,kas42 both have church service in the native language4 ,welsh1 #bjj1 in ,rosedale1 and ,irish about midway in ,wyandotte township4 ,,summaries4 ,there are ,german settlements of thirty or more persons in the following counties3 ,allen1 ,anderson1 ,butler1 ,chase1 ,chautauqua1 ,cherokee1 ,cheyenne1 ,coffey1 ,comanche1 ,cowley1 ,crawford1 ,decatur1 ,dickinson1 ,doniphan1 ,douglas1 ,edwards1 ,elk1 ,ellis1 ,ellsworth1 ,ford1 ,garfield1 ,geary1 ,greenwood1 ,harper1 ,harvey1 ,hodgeman1 ,jefferson1 ,kingman1 ,leavenworth1 ,lincoln1 ,marion1 ,marshall1 ,miami1 ,mitchell1 ,montgomery1 ,mc,pherson1 ,nemaha1 ,neosho1 ,norton1 ,osage1 ,osborne1 ,phillips1 ,pottawatomie1 ,rawlins1 ,reno1 ,rice1 ,riley1 ,rooks1 ,rush1 ,saline1 ,sedgwick1 ,shawnee1 ,sherman1 ,smith1 ,stafford1 ,stanton1 ,thomas1 ,wabaunsee1 ,washington1 ,wyandotte4 ,total1 #fj4 ,scandinavians in settlements of thirty or over are found in3 ,allen1 ,chautauqua1 ,cherokee1 ,cheyenne1 ,cloud1 ,cowley1 ,crawford1 ,decatur1 ,dickinson1 ,doniphan1 ,edwards1 ,elk1 ,gove1 ,greeley1 ,greenwood1 ,hodgeman1 ,jackson1 ,jewell1 ,labette1 ,lincoln1 ,logan1 ,lyon1 ,marshall1 ,morris1 ,mc,pherson1 ,neosho1 ,osage1 ,ottawa1 ,pawnee1 ,phillips1 ,pottawatomie1 ,rawlins1 ,riley1 ,saline1 ,sedgwick1 ,sherman1 ,wabaunsee1 ,wallace1 ,wilson1 ,wyandotte4 ,total1 #dj4 ,settlements of ,slavonic peoples1 ,bohemians1 ,poles1 ,russians1 or ,hungarians1 in3 ,decatur1 ,ellsworth1 ,harper1 ,lincoln1 ,marshall1 ,ottawa1 ,phillips1 ,rawlins1 ,reno1 ,riley1 ,rooks1 ,rush1 ,sedgwick1 ,washington4 ,total1 #ad4 ,settlements of ,irish have been made in3 ,anderson1 ,cloud1 ,crawford1 ,dickinson1 ,doniphan1 ,elk1 ,geary1 ,jackson1 ,kingman1 ,marshall1 ,miami1 ,nemaha1 ,osage1 ,ottawa1 ,pottawatomie1 ,washington1 ,wyandotte4 ,total1 #ag4 ,french are found in settlements of thirty or more in3 ,cherokee1 ,cloud1 ,crawford1 ,doniphan1 ,graham1 ,harper1 ,harvey1 ,nemaha1 ,osage1 ,pottawatomie1 ,rooks1 ,sedgwick1 ,washington4 ,total1 #ac4 ,italians are in ,cherokee1 ,crawford1 ,sedgwick4 ,total1 #c4 ,welsh in ,lyon1 ,osage and ,wyandotte4 ,total1 #c4 ,dutch in ,phillips1 ,reno1 ,sedgwick4 ,total1 #c4 ,scotch are reported from ,cherokee and ,osage4 ,total1 #b4 ,english in ,geary and ,doniphan4 ,total1 #b4 ,the following counties report that there are no settlements of people of foreign birth within their borders3 ,atchison1 ,barber1 ,bourbon1 ,clarke1 ,finney1 ,grant1 ,gray1 ,hamilton1 ,haskell1 ,lane1 ,linn1 ,morton1 ,pratt1 ,seward4 ,total1 #ad4 ,no reports have been secured from the following counties3 ,barton1 ,brown1 ,clay1 ,franklin1 ,johnson1 ,kearney1 ,kiowa1 ,meade1 ,ness1 ,republic1 ,russell1 ,scott1 ,sheridan1 ,stevens1 ,sumner1 ,trego1 ,wichita1 ,woodson4 ,total1 #ah4 ,seventy-four of our ,kansas counties report settlements of citizens of foreign birth in numbers above #cj4 ,in so many cases there is no report or estimate of numbers that it is not worth while to give summaries4 ,probably there are not actually ten counties that have not such settlements4 ,church services in a foreign tongue are held as follows3 ,allen ,s41#d ,anderson ,g41 ,butler ,g41 ,chase ,g41 ,cheyenne ,g41 ,cherokee ,g41 ,cloud ,f4 ,s41 ,coffey ,g41 ,decatur ,g41 ,dickinson ,g4 ,s41 ,doniphan ,g41 ,douglas ,g41 ,edwards ,g4 ,s41 ,ellis ,g41 ,ellsworth ,g41 ,ford ,g41 ,geary ,g41 ,graham ,f41 ,greeley ,s41 ,greenwood ,g4 ,s41 ,harper ,g4 ,hung41 ,harvey ,g41 ,hodgeman ,g41 ,jefferson ,g41 ,leavenworth ,g41 ,lincoln ,g4 ,du41 ,logan ,s41 ,lyon ,w4 ,g41 ,marion ,g4 ,boh41 ,marshall ,g41 ,miami ,g41 ,mitchell ,g41 ,montgomery ,g41 ,morris ,s41 ,mc,pherson ,s4 ,g41 ,nemaha ,g41 ,neosho ,g4 ,s41 ,norton ,g41 ,osage ,s4 ,welsh1 ,osborne ,g41 ,pawnee ,s41 ,phillips ,g4 ,du41 ,pottawatomie ,g4 ,s41 ,rawlins ,g41 ,reno ,g4 ,du4 ,rus41 ,rice ,g41 ,riley ,s41 ,rooks ,f4 ,g41 ,rush ,g41 ,saline ,g4 ,s41 ,sedgwick ,g41 ,sherman ,g4 ,s41 ,smith ,g4 ,du41 ,stafford ,g41 ,wabaunsee ,g41 ,wallace ,s41 ,washington ,g4 ,wilson ,s41 ,wyandotte ,g4 ,s4 ,total1 #eh4 ,this total of fifty-eight counties in which church service is held in a foreign tongue does not at all indicate the number of such churches4 ,in many of the reports received the number is not given1 or merely in the plural4 ,these very incomplete reports indicate one hundred thirty-eight such churches2 it is safe to say that the number is nearly double this4 ,more interesting is the number of schools conducted in a foreign tongue4 ,the counties having them are3 ,allen ,s41 ,anderson ,g41 ,chase ,g41 ,cheyenne ,g41 ,cherokee1 ,g41 ,cloud ,f41 ,dickinson ,g4 ,s41 ,douglas ,g41 ,ellis ,g41 ,ellsworth ,g41 ,ford ,g41 ,geary ,g41 ,greeley ,s41 ,harvey ,g41 ,leavenworth ,g41 ,lincoln ,g4 ,s41 ,logan ,s41 ,marion ,g41 ,marshall ,g41 ,mitchell ,g41 ,morris ,s41 ,mc,pherson ,s4 ,g41 ,nemaha ,g41 ,osborne ,g41 ,phillips ,g41 ,pottawatomie ,g4 ,s41 ,rawlins ,g41 ,reno ,g4 ,du4 ,rus41 ,riley ,s41 ,rush ,g41 ,saline ,s41 ,sedgwick ,g41 ,sherman ,g4 ,s41 ,smith ,g4 ,du41 ,wabaunsee ,g41 ,wallace ,s41 ,washington ,g4 ,total1 #cg4 ,the number of separate schools in a foreign language so far as reported is seventy-four1 and here1 too1 it is safe to say that the actual number is much larger4 ,,explanation4 ,the spaces indicating settlements are in many cases too small to admit a complete description of the inhabitants1 and accordingly they have been marked by races rather than by nationalities and tribes4 8,german0 is made to do duty for all inhabitants of ,germany whether ,low or ,high1 as well as for ,austrians1 ,german ,swiss1 and ,russo-,german ,mennonites4 ,the last are reported simply as ,mennonites1 but are1 ,i believe1 in all cases of ,german origin4 8,scandinavian0 is used instead of ,swede1 ,norwegian and ,dane1 because in some cases the distinction was not made in the reports1 and in order to limit the number of colors on the map which is to come4 ,in the case of ,scotch ,i have been unable to secure information whether they are ,highlanders or ,lowlanders1 and in case of ,irish1 to what extent1 if at all1 they speak the old ,irish language4 ,w4 ,h4 ,,carruth4 ,illustration3 ,a ,,preliminary ,,map ,,of ,,foreign ,,settlements ,,in ,,kansas4 ,b - ,bohemians 7in a few cases other ,slavs7 ,g - ,germans 7including ,dutch and ,russian ,mennonites7 ,s - ,scandinavians 7,danes1 ,swedes and ,norwegians7 ,i - ,irish ,w - ,welsh ,it - ,italians ,f - ,french ,h - ,hungarians ,footnote3 #d ,g ,german1 ,s ,scandinavian1 ,f ,french1 ,w ,welsh1 ,du ,dutch4 ,the ,great ,spirit ,spring ,mound4 ,,by ,e4 ,h4 ,s4 ,,bailey4 ,the 8,waconda0 or ,great ,spirit ,spring1 which is situated in ,mitchell ,county1 ,kansas1 about two miles east of ,cawker ,city1 has been described in detail by ,g4 ,e4 ,patrick in vol4 vii1 p4 #bb1 ,transactions of the ,kansas ,academy of ,science4 ,an analysis of the water1 and of the rock forming the mound on which the spring is located1 is also given4 ,the spring is upon a conical1 limestone mound #db feet in height1 and #aej feet in diameter at the top4 ,the pool itself is a nearly circular lake about #ej feet in diameter1 #ce feet deep1 and the water rises to within a few inches of the top of the basin4 ,there is a level space on all sides of the spring so wide that a carriage can be readily driven around it4 ,there is but little indication of organic matter in the water of the large spring1 though there is a slimy white deposit adhering to the bottom and sides1 but the water is colorless1 clear1 and transparent4 ,the excess of water1 instead of overflowing the bank1 escapes by numerous small fissures1 from #aj to #bj feet down on the sides1 especially on the side away from the bluff4 ,in these lateral springs there is an abundance of green alg1 and a whitish scum1 which seems to be detached from the bottom and to float to the surface4 ,this has a slimy1 granular feeling suggesting in a very marked manner hydrated silica4 ,the mound is situated within about #bjj feet of a limestone bluff1 which rises perhaps #bj feet above the level of the spring4 ,the natural inference would be that the harder material of the mound protected it from the erosion which carried away the rock in the valley of the ,solomon on the south1 and the rock between the spring and the bluff4 ,is it not possible however that the mound has been really made by the successive deposits from the spring8 ,although the mound is plainly stratified1 this need not interfere with the theory1 for the water may have been intermittent in its flow4 ,the rock is very porous1 and on being ground to a thin section is shown to be concretionary in structure4 ,an analysis of the water of the spring 7loc4 cit47 showed that it contained over #aabj grains of mineral matter per gallon1 of which #gge grains were sodium chloride and #bjf grains sodium sulphate1 with #ff grains of magnesium sulphate1 #da grains of magnesium carbonate1 and #ca grains of calcium carbonate4 ,an analysis by the author shows that there are #j4hgd grains of silica4 ,samples of the rock composing the mound1 and of the adjoining bluff were secured1 and comparative analyses made1 with the following results3 ,,great ,,country ,,spirit ,,rock4 ,,mound4 ,silica and insoluble residue #b4ad #d4aj ,oxides of ,iron and ,alumina #c4bb #eb4ff ,sulphuric ,anhydride 4#jj #j4cd ,carbon ,dioxide #dj4ij #ci4aj ,calcium ,oxide #ea4ij #di4bh ,magnesium ,oxide #j4fc #a4ae ,water and organic matter1 undetermined #a4ba #fc4cg ------------ ------------ #ajj4jj #ajj4jj ,specific gravity #b4eb #b4gi ,the rocks are entirely different in appearance and structure1 that of the mound being twice as hard as that of the bluff4 ,the former contains much organic matter as is shown by blackening when it is heated in a tube and giving off the characteristic odor4 ,the iron is practically of the ferrous variety1 probably combined with carbonic acid1 and the rock contains traces of chlorides4 ,the particular sample taken was at some distance from the spring1 and had been thoroughly exposed to the weather4 ,the rock of the mound is of just such a character as might have been built up by deposition from the water1 as it contains the least soluble constituents of the water4 ,the process of solidification would have been assisted by the silica in the water1 forming insoluble cementing silicates1 as noticed by ,prof4 ,patrick4 ,the analysis given above shows that there is abundant silica in the water for this purpose4 ,mention has been made of the organic growth in the adjacent springs4 ,the mixed scum on being heated changes from a dull green to a vivid grass-green1 and if ignited it swells up and emits an ill-smelling vapor1 which is evidently nitrogenous in its character4 ,a grayish white ash is left1 which contains much carbonate of lime4 ,this is evidently freshly deposited1 as it is entangled in the alg in granular lumps4 ,a specimen of the white scum1 noticed above1 only slightly mixed with the green alg1 was analyzed4 ,the acid solution of the ash contains #a4bf per cent of soluble silica4 ,this was of course a combined silica1 probably calcium silicate1 which becomes the cementing material in the rock4 ,in another sample of ash1 after removing all the substances soluble in hot water1 the residue was found to contain #gf4df per cent of silica4 ,the siliceous residue from the scum was examined by ,dr4 ,s4 ,w4 ,williston4 ,it consists mostly of diatoms4 ,he recognized ,navicula-- #b species ,nitzschia-- #b species ,asteronella-- #a species4 ,all three genera are found both in fresh and salt or brackish water4 ,the green material consists essentially of ,oscillaria and ,conferv4 ,if the scum is allowed to stand for a short time a very strong sulphuretted odor is developed1 strangely suggestive of salt water marshes or mud flats2 and indeed the same odor is noticed in the vicinity of the spring4 ,no characteristic salt water organisms1 that should occasion this peculiar odor have1 however1 yet been observed here4 ,a more extended and special study of the organic life of these interior salt water marshes and springs would be of great interest4 ,footnotes3 #e ,mostly ,fe,o1 and so calculated4 #f ,with alkalies4 ,on ,pascal's ,limaon and the ,cardioid4 ,,by ,h4 ,c4 ,,riggs4 ,the inverse of a conic with respect to a focus is a curve called ,pascal's ,limaon4 ,from the polar equation of a conic1 the focus being the pole1 it is evident that the polar equation of the limaon may be written in the form3 e #a r ------ cosx ------ 2 p p where e and p are constants1 being respectively the eccentricity and semi-latus rectum of the conic4 ,from the above equation it is readily seen that the curve may be traced by drawing from a fixed point ,o on a circle any number of chords and laying off a constant length on each of these lines1 measured from the circumference of the circle4 ,the point ,o is the node of the limaon2 and the fixed circle1 which ,i shall call the base circle1 is the inverse of the directrix of the conic4 ,this is readily shown as follows3--the polar equation of the directrix is r p_/7e cosx74 ,hence the equation of its inverse is r 7e cosx7_/p1 which is the equation of the base circle of the limaon4 ,if the conic which we invert be an ellipse1 the point ,o will be an acnode on the ,limaon2 if the conic be a hyperbola1 the point ,o is a crunode4 ,if the conic be a parabola1 ,o is then a cusp and the inverse curve is called the ,cardioid4 ,the limaon may also be traced as a roulette4 ,let the circle ,a ,c have a diameter just twice that of the circle ,a ,b4 ,then a given diameter of ,a ,c will always pass through a fixed point ,q on the circle ,a ,b1 7,williamson's ,diff4 ,cal4 ,art4 #bhf7 and will have its middle point on the circle ,a ,b4 ,now any point ,p on the diameter of ,a ,c will always be at a fixed distance from ,c and will therefore describe a limaon of which ,a ,b will be the base circle4 ,the pedal of a circle with respect to any point is a limaon4 ,this may be inferred from the general theorem that the pedal of a curve is the inverse of its polar reciprocal1 7,salmon's ,h4 ,p4 ,c4 ,art4 #abb74 ,for the polar reciprocal of a conic from its focus is a circle and hence its pedal is a limaon4 ,the base circle is the locus of the instantaneous centre for all points on the limaon4 ,let ,b ,o ,p be a line cutting a circle in ,b and ,q4 ,let the line revolve about ,b1 ,q following the circle2 the point ,p will trace a limaon4 ,now1 for any instant1 the instantaneous center will be the same whether ,q be following the circle or the tangent at the point where the line cuts the circle4 ,therefore the instantaneous center for the point ,p is found by erecting a perpendicular to the line ,p ,b1 through ,b1 and a normal to the circle at ,q4 7,williamson's ,diff4 ,cal4 ,art4 #bid74 ,the intersection 7,c7 of these two lines is the instantaneous center for the curve at the point ,p4 ,but by elementary geometry ,c is on the circle4 ,now as the line ,p ,b revolves through #cfj around ,b1 the line ,b ,c which is always perpendicular to it also makes a complete revolution and the instantaneous center ,c moves once round the base circle4 ,below we give a list of theorems obtained by inverting the corresponding theorems respecting a conic4 ,in these theorems any circle through the pole is called a nodal circle1 any chord through the pole is called a nodal chord1 and the line through the pole perpendicular to the axis of the curve is called the latus rectum4 ,the letters e and p signify respectively the eccentricity and half the latus rectum of the inverted conic4 ,the locus of the point of intersection of two tangents to a parabola which cut one another at a constant angle is a hyperbola having the same focus and directrix as the original parabola4 ,the locus of the point of intersection of two nodal tangent circles to a cardioid which cut each other at a constant angle is a limaon having the same double point and director circle4 ,the sum of the reciprocals of two focal chords of a conic at right angles to each other is constant4 ,the sum of any two nodal chords of a limaon at right angles to each other is constant4 ,p ,q is a chord of a conic which subtends a right angle at the focus4 ,the locus of the pole of ,p ,q and the locus enveloped by ,p ,q are each conics whose latera recta are to that of the original conic as #b 3 #a and #a 3 #b respectively4 ,if ,p and ,q be two points on a limaon such that they intercept a right angle at the node1 then the locus of the point of intersection of the two nodal circles tangent at ,p and ,q respectively1 is a limaon whose latus rectum is to that of the original limaon as #b 3 #a4 ,and the envelope of the circle described on ,p ,q as a diameter is a limaon1 whose latus rectum is to that of the original limaon as #a 3 #b4 ,if two conics have a common focus1 two of their common chords will pass through the point of intersection of their directrices4 ,if two limaons have a common node1 two nodal circles passing each through two points of intersection of the limaons1 will pass through the point of intersection of their base circles4 ,two conics have a common focus about which one of them is turned2 two of their common chords will touch conics having the fixed focus for focus4 ,two limaons have a common node about which one of them is turned2 two of the nodal circles through two of their points of intersection will envelope limaons having fixed node for node4 ,two conics are described having the same focus1 and the distance of this focus from the corresponding directrix of each is the same2 if the conics touch one another1 then twice the sine of half the angle between the transverse axes is equal to the difference of the reciprocals of the eccentricities4 ,if two limaons are described having the same node and base circles of the same diameter1 and if the limaons touch each other1 then twice the sine of half the angle between the axes of the limaons is equal to the difference of the eccentricities4 ,if a circle of a given radius pass through the focus 7,s7 of a given conic and cut the conic in the points ,a1 ,b1 ,c1 and ,d2 then ,,sa4 ,,sb4 ,,sc4 ,,sd is constant4 ,if a circle of a given radius pass through the node 7,s7 of a given limaon and cut it in ,a1 ,b1 ,c1 and ,d2 then #a ------------------------------ is constant4 7,,sa4 ,,sb4 ,,sc4 ,,sd7 ,a circle passes through the focus of a conic whose latus rectum is #bl and meets the conic in four points whose distance from the focus are r1 r1 r1 r1 then #a #a #a #a #b ------ ------ ------ ------ ------ 4 r r r r l ,a circle passes through the node of a limaon whose latus rectum is #bl1 meeting the curve in four points whose distances from the node are r1 r1 r1 r1 then r r r r #bl4 ,two points ,p and ,q are taken1 one on each of two conics which have a common focus and their axes in the same direction1 such that ,,ps and ,,qs are at right angles1 ,s being the common focus4 ,then the tangents at ,p and ,q meet on a conic the square of whose eccentricity is equal to the sum of the squares of the eccentricities of the original conics4 ,two points ,p and ,q are taken one on each of two limaons which have a common node and their axes in the same direction1 such that ,,ps and ,,qs are at right angles1 ,s being the common node4 ,then the nodal tangent circles at ,p and ,q intersect on a limaon the square of whose eccentricity is equal to the sum of the squares of the eccentricities of the original limaons4 ,a series of conics are described with a common latus rectum2 the locus of points upon them at which the perpendicular from the focus on the tangent is equal to the semi-latus rectum is given by the equation p -r cos #bx ,if a series of limaons are described with the same latus rectum1 the locus of points upon them at which the diameter of the nodal tangent circle is equal to the semi-latus rectum1 is given by the equation pr -cos #bx ,if ,,pop be a chord of a conic through a fixed point ,o1 then will tan ,p,,so tan ,,pso be a constant1 ,s being the focus of the conic4 ,if ,,pop be a nodal circle of a limaon passing through a fixed point ,o1 then will tan ,p,,so tan ,,pso be a constant1 ,s being the node4 ,conics are described with equal latera recta and a common focus4 ,also the corresponding directrices envelop a fixed confocal conic4 ,then these conics all touch two fixed conics1 the reciprocals of whose latera recta are the sum and difference respectively of those of the variable conic and their fixed confocal1 and which have the same directrix as the fixed confocal4 ,limaons are described with equal latera recta and a common node4 ,also the director circles envelop a fixed limaon having a common node4 ,then these limaons all touch two fixed limaons whose latera recta are the sum and difference respectively of the reciprocals of the variable limaon and of the fixed limaon1 and which have the same base circle as the fixed limaon4 ,every focal chord of a conic is cut harmonically by the curve1 the focus1 and the directrix4 ,every nodal chord of a limaon is bisected by the base circle4 ,the envelope of circles on the focal radii of a conic as diameters is the auxiliary circle4 ,the envelope of the perpendiculars at the extremities of the nodal radii of a limaon is a circle having for the diameter the axis of the limaon4 ,below we give a number of theorems respecting the cardioid obtained by inverting the corresponding theorems concerning the parabola4 ,the straight line which bisects the angle contained by two lines drawn from the same point in a parabola1 the one to the focus1 the other perpendicular to the directrix1 is a tangent to the parabola at that point4 ,the nodal circle which bisects the angle between the line drawn from any point on a cardioid to the cusp and the nodal circle through the point which cuts the director circle orthogonally1 is a tangent circle at that point4 ,the latus rectum of a parabola is equal to four times the distance from the focus to the vertex4 ,the latus rectum of a cardioid is equal to its length on the axis4 ,if a tangent to a parabola cut the axis produced1 the points of contact and of intersection are equally distant from the focus4 ,if a nodal tangent circle cut the axis of a cardioid1 the points of intersection and of tangency are equally distant from the cusp4 ,if a perpendicular be drawn from the focus to any tangent to a parabola1 the point of intersection will be on the vertical tangent4 ,if a nodal circle be drawn tangent to a cardioid1 the diameter of such circle passing through the cusp will be a common chord of this circle and another described on the axis of the cardioid as diameter4 ,the directrix of a parabola is the locus of the intersection of tangents that cut at right angles4 ,the base circle is the locus of the intersection of nodal circles tangent to a cardioid1 which cut orthogonally4 ,the circle described on any focal chord of a parabola as diameter will touch the directrix4 ,the circle described an any nodal chord of a cardioid as diameter will be tangent to the base circle4 ,the locus of a point from which two normals to a parabola can be drawn making complementary angles with the axis1 is a parabola4 ,the locus of the point through which two nodal circles1 cutting a cardioid orthogonally1 and making complementary angles with the axis1 can be drawn is a cardioid4 ,two tangents to a parabola which make equal angles with the axis and directrix respectively1 but are not at right angles1 meet on the latus rectum4 ,two nodal circles tangent to a cardioid which make equal angles with the axis and latus rectum1 respectively but do not cut orthogonally intersect on the latus rectum4 ,the circle which circumscribes the triangle formed by three tangents to a parabola passes through the focus4 ,if three nodal circles be drawn tangent to a cardioid1 the three points of intersection of these three circles are on a straight line4 ,if the two normals drawn to a parabola from a point ,p make equal angles with a straight line1 the focus of ,p is a parabola4 ,if the two nodal circles cutting a cardioid orthogonally and pass through the point ,p1 make equal angles with a fixed nodal circle1 the locus of ,p is a cardioid4 ,any two parabolas which have a common focus and their axes in opposite directions intersect at right angles4 ,any two cardioids which have a common cusp and their axes in opposite directions intersect at right angles4 ,a number of other theorems on the limaon and cardioid are given in ,professor ,newson's article in this number of the ,,quarterly1 and these need not be repeated here4 ,dialect ,word-,list4 ,,by ,w4 ,h4 ,,carruth4 ,the following are some of the dialect words that have come to one observer's ears within the past triennium4 ,they are all from ,kansas1 unless otherwise noted4 ,they are printed here to interest others1 and to secure a basis for observation4 ,the writer will be under obligations to any one who will note his familiarity with any of these words1 insert others1 or other meanings1 and send them1 with a statement of his place of birth and childhood1 to him at ,lawrence3 among3 all of1 as1 ,where are you going among you8 all3 all gone1 as1 ,the corn is all4 7,indiana1 ,penn47 ,comp4 ,german4 bat3 a #8hard case4' bid3 in1 to bid the time of day4 7,indiana47 beeslings3 preparation of artificially curdled milk4 7,indiana and ,kansas47 become3 to look well in1 as1 ,he becomes that coat4 bad3 desperate1 as in1 ,a bad citizen a desperate fellow4 behave3 to behave well1 as in1 ,do behave now6 bump on a log3 something lifeless1 as1 ,he sat there like a bump on a log4 bier3 sham1 as in pillow-bier4 7,vermont47 branch3 a small stream4 breeze3 a torrent of talk1 as in1 ,he gave me a breeze4 boo3 dried mucous4 buckle down3 to work persistently4 conniptions3 a fit1 also #8conniption fit4' caba3 an old valise4 7,penn47 craps3 a game with dice2 playing1 it is called1 #8shooting craps4' crawl3 to try to escape from an embarrassing situation without admitting one's mistake4 crawfish3 same as #8crawl4' crock3 an earthenware vessel1 a large bowl4 chuck3 lunch4 chuck-a-luck3 loaded 7of dice74 coddy3 odd1 out of fashion4 chug3 to strike a blow1 as in1 ,chug him one4 could3 to be able1 as in1 ,he used to could4 cod3 a bit of deceit1 as in1 ,he gave the teacher a cod4 ,chenuk3 a ,canadian4 7,note the pronunciation47 dast3 to dare1 as in1 ,he don't dast to do it4 dew-claws3 hands and knees 7871 as in1 ,get down on your d41 apply yourself intensely4 ,dick's hatband3 in the phrase1 ,as contrary as ,dick's hatband4 ,what is the origin of this8 dick-nailer3 anything quite satisfactory1 as in1 ,he 7it7 is a dick-nailer4 drop3 advantage1 as in1 to get the drop on a person 7allusion to dexterity in drawing a revolver74 ,comp4 also3 bulge1 inside-track1 whip handle1 dead-wood1 all used in the same way and with same sense4 diven1 past participle of dive4 drug3 pret4 of drag4 east3 yeast4 emptings3 bread dough set to ferment4 ,note the expression 8,it will come out all right in the emptings10 i4 e4 after it has had a chance to stand4 fat up3 to increase a stake at cards4 find3 to supply with board1 as in1 ,pay five dollars a week and find him2 ,i get five dollars and found4 fresh3 impudent1 7due to greenness74 fog3 to filch4 fluke3 to steal4 7,indiana74 flat3 plug tobacco4 7,arkansas74 gallery3 church1 as in1 ,he's in the gallery4 gag3 an improbable story intended to deceive1 as in1 ,he tried to give me a gag4 go with3 to become of1 as in1 ,what has gone with my hat8 ,ohio2 also in ,pall ,mall ,gazette4 grub-stake3 to give board4 girling3 a #8girl-boy1' in contempt4 gaumy3 not neat4 7,arkansas47 gob1 or gaub3 a shapeless mass1 as1 a gob of mud1 then sportively1 gaubs of wisdom4 gray3 an awkward fellow4 get to3 to get an opportunity to1 as in1 ,he didn't get to do it4 go to3 to intend to1 as in1 ,i didn't go 7for7 to do it4 gumbo3 a peculiar1 putty-like dark soil4 7,kansas47 hen3 feminine1 as in1 hen-party2 comp4 stag-party1 a gathering of men only4 honey3 a fine fellow1 generally ironically4 hump3 to bestir1 as1 ,hump yourself4 hole3 bad condition financially2 as1 ,he is in the hole1 i4 e4 he has lost4 huckleberry3 indifferent1 in1 a huckleberry ,christian4 huckleberry3 the right person1 as in1 ,you're my huckleberry4 hornswoggle3 to discomfit1 as in1 ,i'll be hornswoggled if ,i'll do it4 in it3 on the successful side1 as1 ,he is not in it1 i4 e41 ,he has no prospect of success4 ,this phrase is universal in #ahia4 infare3 the reception after a wedding4 in3 on the credit side1 as1 ,i was in five dollars4 jay3 a green1 conceited fellow4 jag3 a bit of anything2 a spree1 a brief drunk4 jack mosquito3 a large insect of the mosquito family1 three times the size of the pestiferous kind2 this one does not bite4 jimmy3 to meddle1 as1 to jimmy with a thing or person1 to #8fool with4' ,comp41 to #8monkey with4' jump3 to leave without notice1 as to jump the town1 to jump bail2 to jump a board-bill is to leave it unpaid4 joint3 an illegal saloon4 ,what is the origin of it8 jigger or chigger3 a minute red mite1 which frequents weeds and lawns1 burrows beneath the human skin and causes excruciating itching4 keep3 board and lodgings1 as1 ,he works for his keep4 lay over3 to surpass1 as1 ,that lays over anything ,i know4 larofamedlers3 a phrase used generally as equivalent to1 ,it's none of your business4 7,maryland1 ,penn41 ,ohio1 ,arkansas47 ,the word is a corruption of ,lay-over for medlers1 a lay-over being a bear-trap consisting of a pit covered with boughs4 light out3 to start on the run1 as1 ,he #8lit' out for home4 lagniap3 the extra in a bargain1 as1 ,five dollars1 and a hat for lagniap4 7,louisiana47 lush3 to drink heavily1 to #8swill4' mog3 to move1 as1 ,mog along with you4 mogle3 the same4 main3 very1 as1 ,it's main strange4 7,worcester county1 ,mass47 mosey43 to move along with a strut4 move3 motion1 as in1 ,get a move on you4 mealer3 one who takes only meals at a boarding-house4 mind off3 to ward off 7flies1 etc474 meet up with3 to meet4 7,tennessee47 peter out3 to dwindle4 pail3 to milk1 as1 to pail the cow4 7,penn47 possessed3 anything1 as1 ,he acted like #8all possessed4' quill3 to write4 7,the ,writer47 quill-wheel3 a #8rattle-trap' wagon4 ruther3 choice1 as1 ,if ,i had my ruther2 also1 druther4 ride and tie3 verbal phrase1 describing a mode of travel in which one vehicle is used by two sets of people1 one riding ahead a given distance and tying the team where the others who have walked will come up to it1 the first walking on ahead until overtaken and passed by the second1 and so on4 7,colorado47 red up3 to make tidy4 ring off3 to desist or cease talking1 technical phrase from the telephone1 but passed into common usage4 ,comp41 8saw off40 rucus3 quarrel1 rumpus4 saddy3 thanks1 thank you4 7,penn47 saw off3 #8ring off2' a short person is said to look #8sawed off4' shet1 shut1 shed3 rid1 as1 to get shut of anything4 shear off3 to pour off 7water from settlings74 7,ohio47 shapin's3 young peas and beans--the unfilled pod4 7,arkansas47 should have said3 said1 as1 ,he should have said yes1 i4 e41 indeed he said yes4 shin3 to climb1 as1 to shin up a tree4 shut off3 to make to stop talking1 as1 ,do shut him off4 shebang3 anything run-down1 as house1 carriage1 affairs4 scrooch or scrooge3 to cringe4 skin-away3 a small boy4 7,civilized ,sac ,indians47 skin3 to run1 as1 ,skin out1 i4 e41 run away4 skid3 to sneak through examinations4 7,yale47 skid3 a sharp-pointed instrument4 skit3 a mild lie4 skads3 great quantities1 as1 ,skads of money1 of books1 etc42 also money1 as1 ,he hasn't the skads4 singed cat3 a shrewd #8rustler1' of unpretentious appearance4 skulduggery3 knavery4 skip3 to run away1 as1 ,now skip1 i4 e41 ,go away from here4 skip3 to leave hastily1 as1 ,he skipped the town4 slouch3 a gawky fellow2 then anything imperfect1 as in the phrase1 ,he's no slouch1 i4 e41 ,he is an expert2 no slouch of a horse1 i4 e41 a first-rate horse4 sloomiky3 not neat4 slander3 to saunter4 slump3 to fail to meet requirements1 as1 in examinations4 slumps3 great quantities4 7,clark's ,second ,hand ,catalogue1 ,n4 ,y47 sleep3 to give lodgings4 ,i have heard1 ,we can eat and sleep him4 smoodle3 a sycophant4 7,kansas ,university47 ,comp41 #8swipe4' smokewood3 dried water-soaked wood used by small boys as substitute for cigars4 smearcase3 a preparation of clabber1 often called #8,dutch cheese4' snake3 to snatch stealthily4 snum3 to vow1 as in1 ,well1 ,i snum4 ,reported as common among girls4 snouge3 unfair1 as1 a snouge game4 snide3 inferior1 unfair1 as1 a snide game1 a snide watch1 etc4 so fashion3 thus1 as1 ,do it so fashion4 soap3 bribe money in elections4 sugar3 same as soap4 sugar1 7explet73 pshaw6 split3 anything1 as1 ,he ran like split1 also1 lickety split4 spunky3 pouting1 incensed4 sprinkle 3 a small number2 also a considerable number4 sprinkling stag3 masculine1 as1 ,a stag-party4 stag3 to go to an entertainment without a lady companion1 as1 to stag it4 ,stoughton-bottle3 an unimpressionable fellow4 7,from ,stoughton's ,bitters1 common in the #ej's47 streak3 rapid rate1 as1 ,he talked a streak1 or more commonly1 a blue streak4 streak3 to run1 as1 ,he streaked it for home4 steer3 to manage 7votes71 as1 ,a steering committee1 the same as #8whips' in ,parliament4 striffin1 or strifning3 the membrane surrounding the abdominal viscera4 7,missouri47 swan3 to vow1 as1 in exclamation1 ,i swan6 swat3 to slap or strike1 as1 ,swat him in the eye4 suz3 7excl473 me1 as in1 ,dear suz1 and ,law suz4 swipe3 a sycophant4 7,harvard47 tacky3 not fashionably dressed4 tewed3 harrassed1 as1 ,i'm tewed and fretted4 that3 so1 as in1 ,not that far4 throw over3 to #8cut' 7an acquaintance74 throw over3 to stop1 as1 ,i threw her over1 i4 e41 stopped talking4 ,common among railroad men2 derived from the use of the reverse lever4 tear-down3 to thrash1 as1 ,he gave the boy a good tearing down4 toad on a tussock3 anything dull or lifeless4 ,he sat there like a toad on a tussock4 tousle3 to disarrange 7hair74 tousey3 frowsy4 topside3 on top of1 as in1 ,the best man topside o' ,god's green earth4 trade-lash3 an exchange of compliments4 7,wellesley47 trottin'-riggin's3 best suit of clothes4 two sticks3 anything1 as1 ,he's as cross as two sticks4 up above3 up1 as in1 ,up above stairs4 whootle-dasher3 a #8rustler4' want3 for was not1 were not1 etc4 wamus1 wampus1 warmus3 a close1 generally knit jacket4 7,illinois1 ,pennsylvania1 ,ohio1 ,wisconsin1 ,new ,england47 ,,prospectus4 ,the ,,kansas ,,university ,,quarterly is maintained by the ,university of ,kansas as a medium for the publication of the results of original research by members of the ,university4 ,papers will be published only upon recommendation by the ,committee of ,publication4 ,contributed articles should be in the hands of the ,committee at least one month prior to the date of publication4 ,a limited number of author's separata will be furnished free to contributors4 ,the ,,quarterly will be issued regularly1 as indicated by its title4 ,each number will contain fifty or more pages of reading matter1 with necessary illustrations4 ,the four numbers of each year will constitute a volume4 ,the price of subscription is two dollars a volume1 single numbers varying in price with cost of publication4 ,exchanges are solicited4 ,communications should be addressed to ,v4 ,l4 ,,kellogg1 ,university of ,kansas1 ,lawrence4