EXCHANGE 
 
MAK 16 1316 
 
 CONTRIBUTIONS TO 
 EQUILONG GEOMETRY 
 
 BY 
 
 PAUL HENRY LINEHAN 
 
 Submitted in Partial Fulfillment of the Requirements for 
 
 THE Degree of Doctor of Philosophy, in the Faculty 
 
 OF Pure Science, Columbia University 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 
 LANCASTER, PA. 
 
 1915 
 
CONTRIBUTIONS TO 
 EQUILONG GEOMETRY 
 
 BT 
 
 PAUL HENRY LINEHAN 
 
 Submitted in Partial Fulfillment of the Requirements for 
 
 THE Degree of Doctor of Philosophy, in the Faculty 
 
 OF Pure Science, Columbia University 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 
 LANCASTER, PA. 
 
 1915 
 
U^' 
 
 § 
 
 a: 
 c 
 
CONTENTS. 
 
 Page 
 
 Introduction v 
 
 Chapter I. 
 
 The Invariants of Irregular Analytic Curves under the Group of Equilong 
 
 Transformations of the Plane 1 
 
 Chapter II, 
 
 Some Curves with Equations in Hessian Line Coordinates. 
 
 Sect. 1. Hessian Line Coordinates 4 
 
 2. Evolutes and Involutes 5 
 
 3. Radii of Curvature 8 
 
 4. Pedals and Negative Pedals 9 
 
 Chapter III. 
 Linear Equilong Transformations. 
 
 Sect. 1. The Integral Linear Transformations 12 
 
 (a) The Operation W = w + ^ 12 
 
 (6) The Operation W = aw 15 
 
 (c) The Operation W = aw + ^ 19 
 
 Sect. 2. The General Linear Transformations 21 
 
 (o) The Operation W = - 21 
 
 w 
 
 (6) Combinations of the Operations W = w + B and W = —. 25 
 
 w 
 
 (c) Combinations of the Operations W = aw ■{- B and W = —. 27 
 
 w 
 
 (d) The Fixed Lines and the Criteria for the Types of Motion. 31 
 
 (e) The Simplest Invariant of a Regular Analytic Curve 35 
 
 3^- 
 
 Si' 
 
 7802 
 
 111 
 
Digitized by the Internet Archive 
 
 in 2008 with funding from 
 
 IVIicrosoft Corporation 
 
 http://www.archive.org/details/contributionstoeOOIinerich 
 
INTRODUCTION. 
 
 The infinite group of equilong transformations of the plane* is that group 
 of contact transformations of the plane by which a straight line becomes a 
 straight line and distance on the line is unchanged. Two arbitrary lineal 
 elements of one and the same line become, under an equilong transformation, 
 two lineal elements of the transformed line, the distance between the points 
 of the original elements being equal to the distance between the points of the 
 new elements. 
 
 If the lines of the plane be determined by the Hessian coordinates {u, v), 
 V being the length of a perpendicular from a fixed point to the line and u being 
 the angle between the perpendicular and a fixed line through the fixed point,t 
 and if the lines of the transformed plane be determined by the Hessian co- 
 ordinates {U, V), the infinite group of equilong transformations will be 
 represented by 
 
 U-\-jV = f(u+jv), 
 
 f{u + jv) denoting an arbitrary analytic function of the so-called dual number 
 u + jv of a system with the unities 1 and j in which j^ = 0. 
 The equilong transformations may be expressed by 
 
 U = <p{u), 
 
 d(p{u) , 
 
 in which (p and x are arbitrary analytic functions of u. 
 
 In this article several topics in equilong geometry are treated. 
 
 Chapter I is concerned with the existence and derivation of the invariants 
 of irregular analytic curves under the group of equilong transformations of 
 the plane. Except for certain types, invariants always exist. The results 
 are roughly analogous to Kasner's results for the conformal group. { 
 
 In Chapter II, the use of Hessian line coordinates in the determination of a 
 lineal element and in the equations of curves is first discussed. Then, by 
 means of these coordinates, the equations of the nth evolute and the nth 
 
 * Cf . Scheffers, Isogonalkurven, Aquitangentialkurven, und komplexe Zahlen, Mathe- 
 matische Annalen, bd. 60. 
 
 t Hessian line coordinates are completely described in Chap. II, 1. 
 
 X Cf. Kasner, Conformal classification of analytic arcs or elements; Poincar^'s local prob- 
 lem of conformal geometry, Trans. Amer. Math. Soc, vol. 16 (1915), pp. 333-349, 
 
VI CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 involute are derived. Expressions for the successive radii of curvature of an 
 analytic curve in terms of the coeflBcients of the infinite series representing the 
 curve are also obtained. Finally it is shown that if an equation be interpreted 
 successively in Hessian line and polar point coordinates the equation represents 
 a curve and its pedal respectively. 
 
 In Chapter III, the equilong transformations of the plane which are repre- 
 sented by the linear functions of the dual variable u + jv are studied. The 
 treatment is largely after the manner of treatment of the conformal trans- 
 formation represented by the linear functions of the complex variable x + iy* 
 It is seen that all linear transformations consist of certain particular trans- 
 formations and their combinations. For the several types of motion, the 
 effect on certain systems of curves, the systems of curves along which the 
 motions may be assumed to take place, the invariants, the fixed lines, and the 
 conditions are obtained. The simplest invariant of the regular analytic curve 
 under the general linear transformation is obtained also. 
 
 The writer acknowledges with great pleasure his indebtedness to Professor 
 Kasner for helpful suggestions and criticisms. 
 
 * Cf . Cole, The Linear Functions of a Complex Variable, Annals of Mathematics, vol. 5. 
 
CHAPTER I. 
 
 THE INVARIANTS OF IRREGULAR ANALYTIC CURVES UNDER THE 
 GROUP OF EQUILONG TRANSFORMATIONS OF THE PLANE. 
 
 If the irregular analytic curve* 
 
 V = agW«/P + a;,+iw(«+i>/P + ag+2W<«+2)/P ^ ... 
 is to go into the irregular analytic curve 
 
 V = A^U^I^ + A^+iU^^+'^lP + A^+^U^^'^lP 4- • • •, 
 
 p and q being positive integers, p > 2 and q > p, under the general equilong 
 transformation 
 
 U = z2 o^ry^, 
 
 r=l 
 
 CO «8 
 
 V =zl vrarvr~^ + zZ bsW, 
 
 r=l «=1 
 
 then the result obtained by eliminating U, V, and v among the four equations 
 must be an identity in u or 
 
 E E ra^5+;fcwf«+('-i>^*^/^ + Z 6.W* - E ^ 
 
 3+n 
 n=0 
 
 ^ + n H-" (1 — r)p 
 
 = 0; 
 
 "^rtl a:!2/!2!... 
 
 X, y, z, • • ' being positive integers or zero satisfying the relation 
 
 x-\-y-\-z-\----^r. 
 
 Suppose p = 2. If the coefficients of any number of the powers of u be 
 each equated to zero, the number of the constants of the transformation 
 present in the equations will always be found to equal or to exceed the number 
 of equations. Hence there are no invariants. 
 
 * An irregular analytic curve (arc) is a curve which, in the neighborhood 'of a given Une 
 taken as the line (0, 0) of the Hessian coordinate system, cannot be represented by setting v 
 equal to a series in integral powers in u but can be represented by a series with fractional 
 exponents. 
 
 1 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 Suppose p > 2. First, if neither q/p nor (q + l)/p be an integer then from 
 the equations formed by equating to zero the coeflScients of w'/p and u^^^^l^, 
 
 aiUq — tti^l^Ag = 0, 
 
 aia^+i - ai<«+«/M5+i = 0, 
 
 the constant oi of the transformation may be ehminated and an invariant 
 
 obtained. 
 
 Secondly, if qfp be an integer, (q -{- l)fp and {q + 2)fp will not be integers. 
 
 From the equations formed by equating to zero the coeflBcients of w(«+i)/i» 
 andi*(«+2)/p^ 
 
 the constant ai may be eliminated and an invariant 
 
 .9+2-p 
 
 «9+r 
 
 obtained. 
 
 «g+2 
 
 9+i-p 
 
 = J 
 
 fl+2 
 
 Fig. 1. 
 
 Thirdly, if {q + l)lp be an integer, q(p and (q + 2)/p will not be integers. 
 Then equating to zero the coeflBcients of u^'^ and u^^^'>Ip gives the two 
 
 equations 
 
 aiaq — ai«/Mg = 0, 
 
 aia^i - ai(«+2)/M,+2 = 0, 
 from which oi may be eliminated and the invariant 
 
 a 
 
 «+2-p 
 
 obtained. 
 
 a 9+2* 
 
 = J 
 
 fl+2 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 3 
 
 Hence every irregular analytic curve, except the curves 
 
 has equilong invariants.* 
 
 Of the types which have no invariants, the form for q = S, 
 
 is represented in Fig. 1. 
 
 * Regular curves are all reducible to the normal form v = and hence have no equilong 
 invariants. Cf. Kasner, Conformal Geometry, Proc. Fifth Inter. Congress of Math., Cam- 
 bridge, 1912, vol. 2, p. 85. 
 
CHAPTER II. 
 SOME CURVES WITH EQUATIONS IN HESSIAN LINE COORDINATES. 
 
 1. Hessian Line Coordinates. 
 
 A plane curve is to be regarded as the envelope of a single infinity of 
 oriented straight lines each of which is determined by the two Hessian co- 
 ordinates (u, v), V being the length of a perpendicular from a fixed point to 
 the line (w, v) and u being the angle between the perpendicular (v) and a 
 fixed line through the fixed point. On the perpendicular (v), the distance 
 from the fixed point to the line {u, v) is considered positive or negative according 
 as the fixed point is on the left or on the right of the direction of the line (w, v). 
 The angle (u) is generated by rotating the fixed line until its positive sense 
 coincides with the positive sense of the perpendicular (v) and is positive or 
 negative according as the rotation is in the counter-clockwise or in the clock- 
 wise direction. 
 
 A lineal element of the plane is ordinarily determined by (x, y, dy/dx), 
 {x, y) being the cartesian coordinates of its point and dyjdx the tangent of the 
 angle which its direction makes with the positive direction of the ic-axis. It 
 may also be determined by the quantities (w, v, dv/du), {u, v) being the Hessian 
 coordinates of its line (direction) and dv/du the distance along its line from 
 the foot of the perpendicular (v) to its point. If (x, y, dy/dx) and (w, v, dv/du) 
 represent the same lineal element, the following relations exist:* 
 
 u = tan ^-3 ^ , 
 
 ax 2 
 
 dy 
 
 ""d^-y 
 
 V = 
 
 Mtr 
 
 
 du 
 
 >Ri)^' 
 
 In general a plane curve is oriented and is assumed to have an equation 
 of the form 
 
 * Cf. Scheffere, loc. cit., p. 520. 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 5 
 
 V = f{u)* 
 
 The equation u = k (a, constant) represents a point at infinity which is, of 
 course, regarded as a curve. 
 
 It is easily seen geometrically that the contact transformation 
 
 U=^ 
 
 u + ^u 
 
 V = 
 
 «, 
 
 dV 
 dU~ 
 
 dv 
 
 is simply rotation of the plane about the fixed point « = through an angle j8i. 
 It is clear also, geometrically, that the transformation 
 
 dV_dv 
 dU~ du' 
 
 is dilatation of the plane through the distance ^. In general, then, a curve, 
 involute of some curve, E, remains an involute of that same curve, E, under 
 the transformation 
 
 V= v-\-^. 
 
 2. E VOLUTES AND INVOLUTES. 
 
 The nth evolute and the nth involute of a curve, v = f(u), may be easily 
 obtained. 
 
 Let the curve v = /(w), the evolute of which is sought, be tangent to the 
 line AB(u, v) (Fig. 2) at B. BD, which is perpendicular to AB, will be tangent 
 to the evolute at D. Let the coordinates of BD be (wi, Vi). Then 
 
 Fig. 2. 
 * This form is easily seen to be similar to the better known "magical equation of the 
 tangent to a curve." Cf. Loria, Sp. Alg. u. Trans. Ebene Kurven (Gr. trans, by Schiitte), 
 bd. II, p. 255. 
 
6 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 0C= AB 
 or 
 
 dv df(u) 
 
 Replacing « by Wi — ^ » 
 
 is obtained. Hence the equation of the first evolute of the curve 
 
 V =fiu) 
 
 IS 
 
 '=/'(«-!)• 
 
 Since, however, the substitution of w + r for u is equivalent simply to 
 
 rotating a curve through the angle — 7r/2 about the point » = of the plane, 
 the equation of the first evolute of the curve 
 
 V = f(u) 
 may be taken to be 
 
 It follows that the nth evolute of a curve 
 
 V = f{u) 
 is 
 
 |,=/(n)(w), 
 
 where 
 
 /("Hw) 
 
 du"* 
 
 To place the evolute in its natural position with respect to the curve, it should 
 be rotated through the angle n7r/2 or its equivalent. 
 
 The equation of the involute of a curve may be deduced independently of 
 the results just obtained. It may also be simply derived by means of those 
 results. If the equation of the involute of the curve 
 
 V = f{u) 
 is assumed to be 
 
 V = g(u), 
 then 
 
 g'{u)=f{u) 
 and 
 
 ^(w) = J f{u)du + constant. 
 
CONTKIBUTIONS TO EQUILONG GEOMETRY. 7 
 
 Hence the equation of a first involute of the curve 
 
 « = f{u) 
 is 
 
 V = J f(u)du + constant. 
 
 The variation of the constant through all possible values gives the oo^ 
 involutes of the curve. In particular, the constant may take the value zero. 
 Consequently it follows that the first involute of the curve ^ 
 
 V = f{u) 
 may be written 
 
 V = J f{u)du 
 and the n involute may be written 
 
 /(n) 
 
 /(w)dw''. 
 
 If the constant is not taken equal to zero for each successive involute, 
 the equation of the nth involute is 
 
 /in) 
 
 the c's being constants. 
 
 To place the nth involute in its natural positive with respect to the curve 
 it should be rotated through the angle Smr/2 or its equivalent. 
 
 The results just obtained may be used to find the involutes of a circle. 
 If the equation of a circle be 
 
 V = a, 
 
 then the equation of the first involute of a circle is 
 
 V = au -\- b 
 and of the second involute 
 
 « = r n^ + 6m + c 
 
 and of the nth involute 
 
 » = ^"" + (S^!"""' + (^!""" +••• + *• 
 Hence an equation of the form 
 
 V = CnW + Cn-iM**"^ + • • • + Cq 
 
 represents the nth involute of the circle 
 
 V = Cn-nl 
 
8 contributions to equilong geometry. 
 
 3. Radii of Curvature. 
 
 The radii of curvature of the successive evolutes of a curve (analytic arc) 
 at a tangent of the curve are expressible conveniently in terms of the coeflS- 
 cients of the infinite series representing the arc, and conversely. Let the 
 radius of curvature of the kth. evolute be denoted by r^, the radius of curvature 
 of the curve being Tq. Since 
 
 ds dH 
 
 then 
 
 _ dfQ _d^s _dv dh 
 
 ^ du du^ du du^ 
 and 
 
 _ dH d^h 
 
 An analytic arc is represented by the series 
 
 U = Co + Ci2* + C2U^ + • • • + CmV^ + * * ', 
 
 the c's being constants. By rotation and translation* the arc may be so 
 placed that the point of tangency is the fixed point and the normal through 
 the point of tangency is the fixed line of the system of coordinates. Accord- 
 ingly, an analytic arc may be represented by the infinite series 
 
 ia = CiV? + CzU^ -\- CiU^ + • • • + CmU"^ + • • • . 
 
 At the line (0, 0), 
 
 dv 
 
 du"- 
 
 -r-5= 2c2, 
 
 and 
 
 Hence 
 
 d'^v 
 dvT 
 
 d^v 
 dh 
 d V d V 
 
 * Translation {X = x + Pi] Y = y + 02', dY/dX = dy/dx) is expressed by the equa- 
 tions U = u; y = t> + ft cos u + ft sin u; dV/dU = dv/du — ft sin u + ft cos u. 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 If these equations be solved for the c's, then 
 
 
 d^l, 
 
 
 
 C2 = 
 
 2! " 
 
 21' 
 
 
 C3 = 
 
 dh 
 
 du^ 
 
 3! 
 
 d'^ 
 
 ~3!' 
 
 
 Ci = 
 
 du* 
 
 4! 
 
 ra - fo 
 41 
 
 » 
 
 Cf, = 
 
 d^v 
 
 dw' 
 
 ' 5! 
 
 rz— Ti 
 51 
 
 » 
 
 d^+h 
 
 p 
 
 E (- 
 
 l)'^V^2-2p 
 
 and 
 
 ^"+'~ (n+2)! (n + 2)I 
 
 in which p takes positive integral values. 
 
 By means of these results, the Cauchy test for the convergeney of series 
 may be interpreted geometrically. If the series 
 
 V = C^W^ + • • • + CkU^ + • • • 
 
 represent an analytic arc, then the superior limit, as 7i = oo , of 
 
 n+2 
 
 p < (nl2) + 1 
 
 i; (- i)^v,H-2-2p 
 p=i 
 
 (n+2)! 
 
 is finite. If this condition be not fulfilled then the series is divergent and 
 represents geometrically not an analytic curve but what Kasner* has called a 
 "divergent differential element of infinite order." 
 
 4. Pedals and Negative Pedals. 
 
 Suppose the equation of an oriented curve be expressed in polar coordinates 
 (p, 6). Let the coordinates be slightly modified by the following conditions. 
 Let the direction on the radius vector (p) from the pole to a point of a curve 
 be positive or negative according as the pole lies on the left or on the right 
 of the positive direction of the curve at the point. Let the conditions which 
 govern the sense of u also govern the sense of the vectorial angle 6. Let the 
 positive direction on the polar subnormal (dp/dd) be that which makes with 
 
 * Cf . Bull, of the Am. Math. Soc, vol. XX, no. 10, p. 531. 
 
10 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 the positive sense of the radius vector (p) an angle of + 7r/2. The three 
 magnitudes (p, 6, dp/dd) may be used to determine a lineal element of the 
 plane. The point of the element is fixed by (p, 6). The direction of the 
 element is determined by dp/dd, being the direction perpendicular to the line 
 (normal of the arc of the curve) joining the point (p, 6) to the extremity 
 (other than the pole) of the polar subnormal and being determined in sense 
 by the convention determining the sense of the radius vector (p). 
 Consider now the transformation 
 
 e = u, 
 
 dp dv 
 dd du' 
 
 This is seen geometrically (Fig. 3) to be the foot point transformation, the 
 element {u, v, dv/du) becoming the element (6, p, dp/dd). The result is ob- 
 tained analytically by expressing the transformation in the coordinates 
 {x, y, dy/dx), representing the lineal element (w, v, dv/du), and {X, Y, dY/dX), 
 representing the lineal element (6, p, dp/dd). In this notation, the transforma- 
 tion is 
 
 dy 
 
 X V 
 
 i ^^ , .,.„ dx ^ ..Y ,dy TT 
 
 VZ2 + y2 = , _ ^^ , tan-i ^ = tan-i:r - - , 
 
 X dx 2 
 
 
 dy 
 dx 
 
 ^§-y 
 
 >R1)^' 
 
 or 
 
 dy f dy \ dy 
 
 GI-) 
 
 x=± ^^ & y j,^^ dx 
 
 ^^m' ^H-dr 
 
 dY "{dxj * ■'^dx 
 
 dX 
 
 m-^^Hty 
 
 which represents the foot point transformation or the foot point transforma- 
 tion followed by rotation through the angle tt, according as the upper or lower 
 signs are taken respectively. 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 11 
 
 Hence if a curve be expressed by the equation v = f{u), in Hessian line 
 coordinates, its pedal, with respect to the point » = 0, is expressed by the 
 same equation p = f{6), in polar point coordinates. Conversely, if a curve 
 
 Fig. 3. 
 
 be expressed by the equation p = f(d), in polar point coordinates, its first 
 negative pedal with respect to the point p = 0, is expressed by the same 
 equation, v = f(u), in Hessian line coordinates. ' 
 
 For example, the equation 
 
 P = Cn^** + Cn^l^"-! + • • • + Co 
 
 represents the pedal of the nth involute (v = c„w** + c„_iw"~i + • • • + Co) 
 of a circle (v = w!c„) with respect to the center of the circle and the equation 
 » = a" represents the first negative pedal of the equiangular spiral (p = a^) 
 with respect to its pole and represents, consequently, an equiangular spiral. 
 
 The beauty of Scheffers' treatment of equilong transformations is due to the combining 
 of the Hessian line coordinates with the dual number u + jv (j^ = 0). If polar coordinates 
 were combined with this dual number, an infinite group of point transformations would 
 be obtained under which the difference of the polar subnormals (with respect to a fixed point) 
 of two concurrent curves at their common point would be invariant. 
 
CHAPTER III. 
 
 LINEAR EQUILONG TRANSFORMATIONS. 
 
 1. The Integral Linear Transformations. 
 
 (a) The Operation W = w + /3. 
 The integral linear function, W = U -\- jV, of the dual variable, 
 
 w = u -\- jv, 
 is 
 
 W = aw + P, 
 
 where a and /3 are dual numbers of the same form as W and w. 
 If a = 1, the function becomes 
 
 W= w + fi, 
 
 which represents the transformation 
 
 U + jV= u + jv + Pi + j^ 
 or 
 
 f^ = w + /3i, 
 
 V = v+^, 
 
 dV_dv^ 
 dU du ' 
 
 First, let /3i 4= and /^ = 0. Replacing the coordinates (w, r, dvjdu = v'), 
 which may be referred to as the Hessian coordinates of a lineal element, by 
 the coordinates (x, y, dyjdx = y'), cartesian coordinates, the transformation 
 takes the form 
 
 tan-ir-|=tan-i2/'-| + ^i, 
 
 XY' - Y xy' -y 
 ^1 + Y'^ ~ ^l + y'^' 
 X+YY' ^ x^yy^ 
 
 or, solving for X, Y, and Y', 
 
 * The equilong transformations may be regarded as transformations of lines or of lineal 
 elements in the plane. 
 
 12 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 13 
 
 X = X COS Pi — y sin fii, 
 Y = X smPi-{- y cos /3i, 
 y' + tan /3i 
 
 Y' = 
 
 1 — y' ' tan /3i * 
 
 These equations define rotation through the angle j8i about the point v = 0. 
 Every curve v == k, a, circle with center at this point, is unchanged as a whole 
 although its elements move in order around it.* The circles v = k are there- 
 fore the curves of motion. f 
 
 Suppose, secondly, that jSi = and P2 4= 0. The transformation becomes 
 
 U=u, 
 
 V = v + ^, 
 
 dU du' 
 
 In cartesian coordinates, it is 
 
 tan ^ y — — = tan y — ^ , 
 
 XY'-Y xy'-y 
 
 ~r P2, 
 
 or 
 
 Vi_|_ y/2 
 
 <\^y"' 
 
 Z+ FF' 
 
 x-\- yy' 
 
 Vi 4- y'2 
 
 Vl + 2,'^' 
 
 Z = a; + 
 
 Piy' 
 
 F = i/- 
 
 /S2 
 
 ^1 + .'^' 
 
 1^' = y'. 
 
 
 which defines dilatation. 
 
 The orientation of the Hessian coordinates shows that the dilatation is in 
 the direction which makes with the positive direction of the lineal element an 
 
 * An equilong transformation always turns every curve m = &, a point at infinity, into 
 a curve « = c, another (or sometimes the same) point at infinity. 
 
 t The particular curve along which a particular line may be regarded as moving may be 
 obtained by substituting the coordinates of the line in the equation of the family of curves, 
 solving for the parameter, and then substituting the value obtained in the equation of the 
 family of curves. If a line move from the position (a, 6) to the position (A, B) along the 
 
 curve V — /(u), the distance passed over is I [/(«) +/"(u)]dw. 
 
14 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 angle of T 7r/2 according as ^ is positive or negative. Each circle, v = k, 
 with center at the point v = 0, becomes another circle, v = k -{• P2, with center 
 at the same point. The curves of motion are the points at infinity, u = k. 
 The transformation 
 
 dU~du' 
 
 in which j3i =|= and 182 4= 0, is rotation and dilatation in succession. Since 
 dv/du is unchanged by the transformation, the 00 1 systems of curves 
 
 dv _ 
 du 
 
 are each unchanged. K the individual curves of one of the systems 
 
 V = CU+ k 
 remain unchanged, then, after the substitution 
 
 u= U-fii, 
 
 every curve 
 
 V= cU-c^i + k + ^ 
 
 must be identical with every curve 
 
 V = cu-\- k. 
 
 If this be so, 
 or 
 
 Hence the curves 
 
 -cPi-\-k-}-^ = k 
 
 ft 
 
 ^ = «"• 
 Pi 
 
 ft I I 
 v = —u+k, 
 
 Pi 
 
 the 00* first involutes of the circle v = ft/ft, are each unchanged as a whole 
 and may be regarded as the curves along which the lines move under the 
 transformation. 
 
 Since dv/du is invariant and since 
 
 d _d_ dU _d^ 
 
 du~ dU' du~ dU ' 
 or 
 
 ±_d_ 
 du~ dU' 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 15 
 
 every derivative d'^vldu^ is invariant. Consequently the radius of curvature 
 of every evolute of a curve is invariant. An involute of a curve is transformed 
 therefore into another involute of the same curve. For rotation, one invariant 
 is 
 
 and, for dilatation, 
 
 dv dh 
 
 du du^ ^ 
 
 which last is also an invariant of the rotation-dilatation transformation 
 
 (&) The Operation W = aw. 
 If, in the integral linear function, i3 = 0, the transformation 
 
 W = U -\- jV = aw = (ai + ja2){u + jv) 
 or 
 
 U = aiU, 
 
 V = aiV + a2U, 
 
 dV _ dv a2 
 dU du ai 
 is obtained. 
 
 First, let 0:2 = 0. The transformation is then 
 
 U = aiU, 
 
 V = aiV, 
 
 dV_di 
 dU du ' 
 
 The line (0, 0) is fixed. A circle v = k, with center at the point v = 0, goes 
 into a circle v = aik, with center at the same point. Since 
 
 every curve 
 
 is unchanged as a whole. Hence the curves of motion are v = Jcu, those 00* 
 first involutes of the 00 ^ circles v = k, which are all tangent to the fixed line 
 (0, 0). 
 
 * It is evident that Hessian line coordinates are convenient for the treatment of rotation 
 and dilatation. 
 
 V 
 
 u 
 
 aiV 
 aiu 
 
 = 
 
 V 
 
 
 V 
 
 u 
 
 k 
 
 
16 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 Secondly, let ai = 1. The transformation is then 
 
 U=u, 
 V = V + ttiU, 
 
 dU~du'^''" 
 
 The points at infinity, u = k, are the curves of motion. 
 
 If, thirdly, ai 4= 1 and a2 4= 0, every circle v = k, with center at the point 
 e = 0, is transformed into a first involute, v = {az/aiju + aik, of the circle 
 
 Let the equation of the curves along which the lines may be regarded as 
 
 moving be 
 
 V = f{u). 
 
 By the transformation this equation must assume the form 
 
 Hence the equations 
 
 V = /(w) 
 and 
 
 aiv + «2W = fipciu) 
 
 must be identical or the function / must satisfy the equation 
 
 «i/(w) + «2W = /(aiw). 
 Differentiation gives 
 
 /'(w)-/'(«iw)+^=0. 
 
 Differentiation a second time gives 
 
 j"{u) - a,j"{a,u) = 0, 
 or 
 
 r'{u) 
 
 and differentiation a third time, 
 
 f"'{u) - <xiY"{cciu) = 0. 
 By eliminating the coefficient ai between the last two equations, the equation 
 
 f'"{u) r"{aiu) 
 
 is obtained. 
 Hence 
 
 S"\u) 
 
 ,.,,, v,» = a constant. 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 17 
 
 The solution of this differential equation, obtained by direct integration three 
 times in succession, is 
 
 /(w) == a(u + 6) log (w + 6) + cw + d. 
 
 The equation of the curves is then 
 
 V = a{u + b) log (w + 6) + cw + d, 
 
 in which the values of a, h, c, and d are to be determined. If this equation is 
 to be unchanged in form by the transformation, then 
 
 Hence the oo ' curves along which the lines may be regarded as moving under 
 the transformation are represented by the equation 
 
 V = — i — i — u • log U-+- cu. 
 ailoglai 
 
 ASYMPTOTE 
 
 Fig. 4. 
 
 Consider the curve (Fig. 4) 
 
 V = 
 
 «2 
 
 ai log ai 
 
 w-log w + cu 
 
 or 
 
 ai log ai » 
 
 It is of spiral shape since v increases as u increases. If 
 
 M = 
 
18 
 then 
 since 
 and 
 
 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 V= 
 
 [w-Iogw]„=o = 
 
 00. 
 
 [-1 = 
 
 The line (0, 0) is an asymptote. 
 
 The first evolute of this logarithmic* curve is 
 
 V = 
 
 «2 
 
 log 
 
 («-^) + (o + ^^), 
 
 Q!i log ai V 2 / \ ai log ai 
 
 which is the first negative pedal with respect to the point « = of the logarith- 
 mic spiral of Varignon, the equation of which is, in polar coordinates, 
 
 ^ = -(^n^)'''«(''-|)-(-"-in^)- 
 
 The second evolute is 
 
 v{u — x) = 
 
 "2 
 
 ai log ai ' 
 which is the first negative pedal of the hyperbolic spiral 
 
 ailogai 
 with respect to its pole. 
 
 From the equations of the transformation, it follows that 
 
 A- A 
 du ~ "' dU 
 
 and hence that 
 
 du- "' 
 
 dU- 
 
 [n = 2, 3, 4, . • .]. 
 
 Therefore every expression of the form 
 
 is invariant, the simplest being 
 
 _du'' _ 
 
 m-l 
 
 ~drv ' 
 _du"*_ 
 
 n-1 
 
 [m = 2, 3, 4, • • • 4= n] 
 
 du^j 
 
 dh 
 
 du^ 
 
 = ^3 
 
 * The curve « = a(« + b) log (u + 6) + cu + d> — o, b, c, and d being constants, — ^will 
 be referred to as a "logarithmic curve." 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 19 
 
 The invariants containing derivatives of even order only may be expressed 
 in terms of the radii of curvature of the curve and its evolutes of even order. 
 Consider an analytic curve in the neighborhood of the line (0, 0). Then 
 
 and 
 
 
 du'-''' 
 
 
 
 dH 
 du' - '' 
 
 - ro, 
 
 
 9=1 
 
 l)'^V2p-2,. 
 
 [p= 1,2,3, •••]. 
 
 Consequently the invariants are 
 
 2p+l 
 
 [^2p+2^ -|2p-l r q=p+\ ~|2p-l' 
 
 The invariant of fourth order 
 
 may be written 
 
 Idu^l 
 
 dS 
 du* 
 
 To' 
 
 r2— tq' 
 
 (c) The Operation W = aw -{- fi. 
 The general integral linear function 
 
 W = aw + fi 
 represents the transformation 
 
 U = aiU-{- jSi, 
 
 V = aiv -j- a^u + jS2, 
 
 dV _ dn az 
 dU du ai ' 
 
 Every circle v = k, with center at the point v = 0, goes into the curve 
 
 «2 , / ^ a2iSi , , \ 
 
 a first involute of the circle v = a2fai. 
 
20 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 . = -,=^ [« - j^] log [« - j4^] + 0« 
 
 By the method employed in the case of the transformation W = aw, the 
 system of curves along which the lines may be regarded as moving is found to 
 be 
 
 ai log ai 
 
 ■ «2/3i + caifii — cjSi — ai^ + ^ 
 
 a system of logarithmic curves with the line 
 
 / /3i (1 -Q!i)/32 4-«2/3i \ 
 
 Vl-ai' (l-«i)' J 
 
 as asymptote.* 
 
 This system of curves may be obtained from the system of curves 
 
 Vl = ^— Ml -log Ml -\- CMi, 
 
 ai log ai 
 for the transformation 
 
 Wi = aiDi, 
 
 by the rotation-dilatation transformation 
 
 w = M?i + ;■ . 
 
 1 — a 
 
 Hence the transformation 
 
 W = aw + fi 
 
 may be regarded geometrically as the transformation 
 
 Wi — aw\, 
 where 
 
 W\ = w — -T- — — 
 1 — a 
 
 and 
 
 1 — a 
 
 or in which the fixed line of the transformation is 
 
 j8 / jSi (l-«i)/32 + «2/gi \ 
 
 l-«°'"Vl-ai* (l-«i)' /• 
 
 The invariants of the transformation expressed by the general integral 
 linear function PF = aw; + j8 are those which are common to the transforma- 
 tions expressed by the particular integral linear functions W = w -{- fi and 
 
 * If ai = 1, the system of curves is 
 and, if /9i =0 also, the system b « = A. 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 21 
 
 W = aw and are, consequently, those of the latter. The invariant of lowest 
 order is 
 
 
 dh 
 
 = Jz. 
 
 = — a 
 
 The system of oo ' logarithmic curves 
 
 Idu'i 
 
 dh 
 du^ 
 or 
 
 V = a{u + 6) log (u -{- b) -{- cu -\- d, 
 
 b, e, and d being arbitrary, is invariant. 
 
 The equilong property of the transformation is easily deduced from 
 
 dV _dv a2 
 dU du ai' 
 
 If two curves, tangent to the line (u, v), have the distances [dv/du]i and [dvldu]^ 
 respectively, their transformed curves, tangent to the line {U, V), will have 
 the distances 
 
 du Ji ai 
 and 
 
 fdVl _ Vd^l 
 ldUii~ldu], 
 
 vdvi _r^] 
 
 2 «1 
 
 respectively, and consequently 
 
 Idui lduii~ldU]2 IdUJi 
 
 and the distance between the points of tangency is invariant. 
 
 2. The General Linear Transformations. 
 
 (a) The Operation PF = — . 
 The reciprocal function 
 
 W = -* 
 
 w 
 
 represents the transformation 
 
 * The lines u + jv where u = are not considered. Throughout the entire article, the 
 concept of a line or of lines at infinity is not considered and division hyj is, of course, eliminated. 
 
22 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 1 u — jv u — jv 
 
 U + jV = 
 
 or 
 
 u -\- jv {u + jv) (u — jv) u^ 
 
 u 
 
 dV _ dv V 
 
 dU du u 
 
 A circle v = k, with center at the point « = 0, is transformed into the 
 curve V = — ku^ which is a second involute of the circle v = — 2k and which 
 is tangent to the line (0, 0). 
 
 Since, from the equations of the transformation, 
 
 the system of oo i curves 
 
 or 
 
 is invariant. Each curve 
 goes into the curve 
 the curve 
 
 do^_v_dV_V 
 du u~ dU U' 
 
 dv ^ _ 7 
 du u 
 
 V = ku- log u -\- cu 
 
 V = ku- log u-\- cu 
 V = ku- log u — cu, 
 
 V = ku- log u 
 
 going into itself. 
 
 Hence the oo ^ logarithmic curves 
 
 V = ku- log u 
 
 are the curves along which the lines may be regarded as moving in order. 
 The transformation 
 
 U + jV = -^ 
 
 u + JV 
 is equivalent to the reverse equilong transformation 
 
 U + jV = ^-^ 
 
 U — JV 
 
 or 
 
 U 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 23 
 
 V 
 
 V = — 
 
 dU~ ~ du^ u' 
 
 followed by the reverse equilong transformation 
 
 U -{- jV = u — jv 
 or 
 
 U=u, 
 
 V = -V, 
 
 dV^ _dv 
 
 dU du ' 
 
 This last transformation is equilong symmetry with respect to the point 
 V = 0. Equilong symmetry with respect to the curve v = f{u) is defined in 
 general by the equations* 
 
 V = -v+ 2/(w), 
 
 dV _ _dv ^dfju) 
 dU du du 
 
 In cartesian coordinates, the transformation in this special case is 
 
 X= -X, 
 
 dY_dy 
 dX dx' 
 
 which is rotation about the point v = through the angle t. Since a lineal 
 element and its symmetrical element are pointed in the same direction, the 
 transformation has the effect of rotating every lineal element through the 
 angle w and then reversing its direction. Every curve is rotated through an 
 angle ir and has its orientation reversed. 
 By the transformation 
 
 u 
 
 V 
 
 V = — 
 
 dV _ dv V 
 dU~ ~du'^u' 
 
 * This definition was given by Kasner, in a course at Columbia University, as a kind of 
 analogue to conformal or Schwarzian symmetry. 
 
24 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 a circle v = k goes into a curve v = ku^, which is a second involute of a circle 
 V = 2k and which is tangent to the line (0, 0). Every curve 
 
 au-\- hv = 0, 
 
 which is a first involute of a circle v = — a/b and which is tangent to the line 
 (0, 0), is invariant. A first involute, not tangent to (0, 0), 
 
 au -\- hv -\- c = 
 
 is transformed into a second involute (spiral of Sturm or Norwich) 
 
 cu^ -j- bv -\- au = 0, 
 
 tangent to (0, 0). A second involute, tangent to (0, 0), 
 
 cu^ -{- bv -{- au = 
 goes into a first involute 
 
 au •\- bv -\- c = 0, 
 
 not tangent to (0, 0). A second involute, not tangent to (0, 0), 
 
 cu^ -\- au •}- bv + d = 
 is transformed into a second involute 
 
 du^ + aw + 6r + c = 0, 
 
 not tangent to (0, 0). Each of the oo^ second involutes 
 
 cu^ + aw + 6« + c = 
 is invariant. 
 
 By the transformation, the two points at infinity w = db 1 are rigid. 
 Every line (± 1, A;) is fixed. A line 
 
 |wl <1. 
 
 V = k 
 becomes a line 
 
 \U\>h 
 
 V>k 
 and a line 
 
 |w|>l, 
 
 V = k 
 becomes a line 
 
 \U\ <1, 
 
 V<k. 
 
 The lines, of course, do not move in order along the curves of any system. 
 
 The transformation is of period 2 and is its own inverse as is also the trans- 
 formation W = Ifw. 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 25 
 
 In the reciprocal transformation, W — Ifw, 
 
 dv ^ _ T 
 du u ^ 
 is invariant. Since 
 
 ""du' ^dU' 
 the invariant of order n may be written 
 
 [ L «w J \_du w J j 
 
 in which the symbol [w(d/dw)]"~^ signifies the repetition (n — 1) times of the 
 operation u{d/du). 
 
 For the reciprocal transformation, the equilong property is easily shown to 
 exist. If (w, V, [dvfdu]i) and (u, v, [dvldu]2) be the coordinates of the lineal 
 elements of two curves tangent to the line (m, v) and if {U, V, [dVldU]i) and 
 {U, V, [dV/dU]2) be the corresponding elements for the transformed curves 
 tangent to the line {U, V), then 
 
 and 
 
 and hence 
 
 IdUJi U~ldui2 u 
 
 r^i _z_ [—1 _- 
 
 ldUJ2 ldUii~lduJ2 Idui 
 
 (6) Combinations of the Operations W = w -\- ^ and W = — . 
 
 If the lines (wo) of a plane move along the curves, first involutes of a circle 
 
 Vo = j82//3i, 
 
 ^ 1 7 
 
 Pi 
 
 the corresponding lines (w = Ijwo) of a second plane, reciprocal to the first, 
 will move along the curves 
 
 V = — ku^ — ttu- 
 Pi 
 
 These are second involutes of the circles v = — 2k and are tangent to the line 
 (0, 0) at the same point {[dvldu]u=o = — P2IP1)' 
 
 The equation of the transformation corresponding to motion along this 
 system of curves may be determined in the following way. Let Wo and w be 
 
26 CONTRIBUTIONS TO EQUILONG GEOMETRT. 
 
 the initial positions and Wq and W the final positions of lines in the first plane 
 and in the reciprocal plane respectively. Since the motion in the first plane 
 corresponds to the transformation 
 
 and since 
 and 
 
 then the motion in the reciprocal plane will be that of the transformation 
 
 or 
 
 w 
 
 Wo== 
 
 Wq-\-^ 
 
 w 
 
 _ 1 
 
 Wo 
 
 W-- 
 
 1 
 
 w 
 
 ^w-\-\ 
 
 If (in /3 = /3i + j^ /3i = 0, the motion is along the points at infinity u = l/k^ 
 which are reciprocal to the points at infinity Uq = k. 
 
 Suppose the system of second involutes to have the line (71, 72) instead of 
 the line (0, 0) in common. The equation will then be 
 
 «> — 72 = — kiu — 7i)2 — — (m — 71). 
 
 Pi 
 
 Subject the plane to the transformation 
 
 Wi = w — y. 
 The equation of the system of second involutes of circles takes the form 
 
 Vi= — ku^ — -^ Ml, 
 
 Pi 
 
 the line (0, 0) being common to all the curves. Since motion along this 
 system corresponds to the transformation 
 
 in which wi and Wi represent respectively the initial and final positions of the 
 
 lines, and since 
 
 wi = w — y 
 and 
 
 Wi=W-y, 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 27 
 
 the transformation 
 
 or 
 
 (1 4- Mw - ^y" 
 
 W = 
 
 Pw- {I- ^y) 
 
 is obtained. This Hnear function then represents motion along a system of 
 curves 
 
 » - 72 = — k{u — 7i)2 — ^{u — 7i), 
 
 Pi 
 
 each curve being a second involute of a circle {v = — 2k) and being tangent to 
 the line (71, 72) at the same point ([dvldu]u=y = — ^l^i). Motion of this 
 kind will be called parabolic. 
 
 If both numerator and denominator of the second member of the equation 
 
 _ (1 + ^y)w - ^7^ 
 ^w + (1 - ^7) 
 
 be multiplied by an arbitrary number 5 and if 
 
 5 + i857 = a' 
 and 
 
 ^5 = 7' 
 and 
 
 6 - j857 = 3', 
 
 then, dropping accents, the transformation will be expressed by the function 
 
 (« - ^Y 
 
 aw ;, 
 
 yw -\- b 
 
 (c) Combinations of the Operations W = aw -\r ^ and W = —. 
 
 In the transformation 
 
 W\ = awi, 
 
 if (in a = ai + jag) (Xz = 0, the curves of motion are Vi = kui, which are 
 first involutes of the circles v = k and which have the line (0, 0) in common. 
 If these curves be subjected to the transformation 
 
 Wq= Wi + 71, 
 
 their equation will take the form 
 
 «o — T/2 = k{uQ — 771), 
 (vu V2) becoming the common hne. If lines (wq) move along this system of 
 
28 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 curves, the reciprocal lines (w = 1/wo) will move along the system of curves 
 
 V = {krji — r}2)u^ — ku. 
 
 Motion along these curves, which are second involutes of the circles 
 
 V = 2(krii — ife) 
 
 and which have the lines (0, 0) and (I/771, — 772/771^) in common, will be called 
 hyperboHc. 
 
 If, in the transformation {Wi = awi), ai = 1, the curves of motion are the 
 points at infinity ui = k. The lines (w = Ifwi) reciprocal to the lines moving 
 along these points will move along the points at infinity u = Ifk. 
 
 For the general transformation 
 
 Wi = awi, 
 
 the curves along which the lines may be regarded as moving are the 00 * 
 logarithmic curves, c being an arbitrary constant, 
 
 Vi = — -, Wi-logMi + CUi. 
 
 ai log ai 
 If these curves be subjected to the transformation 
 
 they become the 00 1 logarithmic curves 
 
 ^0- V2= ^ 1 „^ (wo - Vi) log {uo - vi) + c(wo - Vi)* 
 Qfi log ai 
 
 the reciprocal curves of which are, since w — 1/wq, the qo^ curves 
 
 V = ^^ I ^^ [Viu"^ - u] [log (1 - riiu) - log u]- cu-\- ^ct/i - rjzW- 
 
 The line (0, 0) is a common asymptote of this family of curves since if 
 
 w = 
 then 
 
 v = 
 and 
 
 LdwJu=o~ 
 
 The line (l/r/i, — rjifrji^), the reciprocal of the asymptote (171, 172) of the system 
 of logarithmic curves, is also a common asymptote of this family of curves. 
 
 Subject the plane to the transformation 
 
 * Motion along this system of curves corresponds to the transformation 
 
 Wo = otWo + i?(l —a). 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 29 
 
 v' = V -\- 72. 
 The asymptote (0, 0) becomes the asymptote (71, 72) and, if 
 
 1 
 
 and 
 
 Vi 
 
 + 7i = 5i 
 
 - -i + 72 = 02, 
 
 the asymptote (Ifrji, — 172/771^) becomes the asymptote (5i, 52). The equation 
 of the system of curves, c being an arbitrary constant, is, dropping accents, 
 
 oi2 r (u — 7i)(m — 
 «i log ai L Ti 
 
 ^^^^ ~ [log (m - 81) - log (w - 7i)] 
 
 , r_ c(7i - 81) + (72 - §2) , (X2 log (71 - 81) "I 2 
 L (71 — 5i)^ «i log «i (71 — 5i) J 
 
 r c(7i — 5i)(7i + 8x) + 271(72 — 82) _ <X2 (71 + ^i) log (71 — 5i) "| 
 L (7i — 5i)^ ai log ai (71 — 61) J 
 
 _l_ r (Ti — SiYjcyi + 72) — C7i^(7i — ^1 
 "^ L (Ti - 5i)' 
 
 ai log ai 
 ) - Ti^(T2 - 82) 
 
 I «2 71^1 log (Ti — 5i) 1 ^ 
 ai log ai (71 — 81) J ■ 
 
 ASYMPTOTE 
 
 ASYMPTOTE 
 
 Fig. 5. 
 
 * A curve with this equation, ai/ai log ai, 71, 72, Si, 82, and c being constants, will be 
 referred to as a curve (C). In Fig. 5 two particular examples of the curve (C) are given. 
 By a system of curves (C) will be meant the « 1 curves (C) obtained by taking definite values 
 for the constants aijai log ai, 71, 72, Si, and S2 and letting the constant c be arbitrary. 
 
30 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 The equation of the transformation represented by motion along this system 
 of curves may be derived in the following way. Suppose the lines are to 
 move from the initial position w to the final position W. If the plane under- 
 go the transformation 
 
 Wo = w — y, 
 
 Wo=W-y, 
 
 the curves, regarded as involutes, will be unchanged but the asymptote 
 
 (Ti, 72) will become the asymptote (0, 0). As lines move along this system of 
 
 curves, with one asymptote (0, 0), from the positions {wq) to the positions 
 
 (Wo), their reciprocal lines {wi = Ijwo) and (JVi = I /Wo) will move according 
 
 to the relation 
 
 Wi = awi + /3. 
 
 Hence 
 
 1 1 ■ o 
 — - = « \-^ 
 
 Wo Wo 
 
 or 
 
 and 
 
 or 
 
 Wo= "° 
 
 W-y = 
 
 j8wo + a 
 w — y 
 
 W = 
 
 p{w — 7) + a 
 
 (1 + Py)w +(ay-fiy^-y) 
 fiw+{a- ^y) 
 
 The asymptote 5 may be introduced instead of the quantity /3. The 
 transformation Wo = w — y changes the line 5 into the line 8 — y. The 
 motion 
 
 Wi = awi + jS 
 
 takes place about the fixed line /3/(l — a) which is the reciprocal of the line 
 5 — 7. Hence 
 
 /3 _ 1 
 1 — a 8 — y 
 or 
 
 1 -a 
 8 — y 
 
 and the transformation takes the form 
 
 ^ (5 - ay)w + icx8y — 8y) 
 (1 —a)w +(a5 — 7) * 
 
 The lines 7 and 5 are the asymptotes of the system of curves (C) and a defines 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 31 
 
 the extent of the motion which is reciprocal to the given motion with respect 
 to the asymptote. 
 
 The transformation may be written 
 
 8(w — y) — ay{w — S) 
 {w — y) — a{w — 5) * 
 
 The reverse transformation is then 
 
 ^ y{ W -8)- a8iW - y) 
 ^ ~ (W- 8)- (x{W - y) ' 
 
 If the first of these moves w to W, the second moves W back to w. Each 
 transformation may be obtained from the other by interchanging y and 5. 
 Hence if the asymptotes of the motion be interchanged, a second motion equal 
 and opposite to the first will be obtained. 
 
 The transformation is the general linear transformation and may be put 
 in the form 
 
 y'w + 5' 
 or, dropping accents, 
 
 aw + j8 
 
 W = 
 
 yw + 5 
 
 {d) The Fixed Lines and the Conditions for the Types of Motion. 
 
 It has been shown that motions along the several systems of curves corre- 
 spond to linear transformations of the variable u + jv. It will now be shown 
 that all linear transformations give rise to the same systems of curves. 
 
 The transformation 
 
 aw + j3 
 
 w ^ 
 
 yw + 5 
 
 in general leaves unchanged two Hues of the plane which are the roots of the 
 equation 
 
 yw^ + (5 — a)w — j8 = 0. 
 
 If the two roots are coincident and infinite, then 
 
 7 = 
 and 
 
 6 - a = 
 
 and the transformation reduces to 
 
 W = W + -. 
 a 
 
 If the transformation is of this form, then it has no fixed lines in the finite plane. 
 
32 CONTWBUnONS TO EQUILONG GEOMETRY. 
 
 If the two roots are coincident and finite, then 
 
 (5 - a)2 + 4/37 = 
 
 and the transformation reduces to 
 
 (8 - a)« 
 
 aw — — -. 
 
 w= ^ 
 
 
 8) 
 
 (a - dy + 4i87 = 
 
 = 
 
 [K« + 8)f 
 
 1 
 
 yw + 5 * 
 
 the motion corresponding to which has been called parabolic. A transforma- 
 tion of this form has two coincident lines fixed in the finite plane. 
 The fixed lines ivi and W2> for parabolic motion, are 
 
 The condition 
 
 may be written 
 
 ! 
 
 a8 — fiy 
 
 If the motion takes place along points at infinity, then, from the equation 
 for parabolic motion in the form 
 
 {\±^y)w±S-M 
 ^w+{i- /St) * 
 
 ^ = + i^2. 
 
 If 
 
 b + ^by = a', 
 
 ^h = 7', 
 and 
 
 b-phy= b\ 
 then 
 
 „ _ 2y' _ (g/ + W)yx' + j[{ax' + 8i072^ - («2^ + 8207/1 
 ^~a' + b' ^ («i' + V)^ 
 
 The further condition that parabolic motion take place along points at in- 
 finity is then, dropping accents, that, in the equation 
 
 (8 - af 
 aw ^, 
 
 W = -rP^— , 71 = 0. 
 
 yw + 
 
 If there is only one fixed line in the finite plane, then 
 
 7 = 0. 
 The transformation is 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 33 
 
 a 8 
 
 W^-^w + l. 
 
 Conversely, this transformation has one fixed line j8/(5 — a), which is the 
 asymptote of the system of logarithmic curves. 
 
 If the two fixed lines are distinct and in the finite plane, one may be made 
 to become the line (0, 0) by the rotation-dilatation transformation, the other 
 becoming at the same time some other line in the plane. By reciprocation, 
 the line not (0, 0) is transformed to some part of the plane. Motion about 
 one line in the finite plane is motion along a system of logarithmic curves. 
 Suppose the operations be reversed. The reciprocal of the motion along the 
 system of logarithmic curves is motion along a system of curves (C) which 
 has the line (0, 0) and the fine into which the second fixed line of the original 
 transformation was transformed as asymptotes. Reversing the rotation- 
 dilatation transformation, motion along the system of curves (C) which has 
 the two distinct fixed lines in the finite plane as asymptotes is obtained. In 
 general, the linear transformation which leaves two distinct lines fixed in the 
 finite plane is represented by motion along a system of curves (C). The 
 fixed lines, Wi and 1^2, of this motion are 
 
 ^(«-5)+V[^(«-5)P4-^7 
 Wi = 
 
 and 
 
 2 
 
 nh = 
 
 Hoc - 5) - V[i(«-5)P + ^7 
 
 To determine the conditions for the hyperbolic case, the equation of the 
 transformation in terms of the fixed lines and the quantity a, now called €, 
 
 _ (vh — iWi)w + WitOije — 1) 
 
 (1 — e)W + €W2 — Wi ' 
 
 is used. If this transformation is identical with the transformation 
 
 aw + /3 
 
 then, fc being a constant. 
 
 Then 
 
 yw -\- 
 
 k(w2 — ewi) = a, 
 
 kwiuhie — 1) = /3, 
 
 kil - 6) = 7, 
 
 k{€W2 — Wi) = 8. 
 
 W2 — €Wi _ a 
 
34 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 and 
 
 , . a + 8 + V(a + 5)2 - 4(a5 - py) 
 
 € = Cl + 762 = , 
 
 a + 5 - V(a + 5)2 - 4(a5 - ^) 
 
 If 
 then 
 
 and 
 
 [{a + 8) + -^(« + 5)2 - 4(a5 - ^y)Y 
 4(a5 - py) 
 
 €2 = 0, 
 
 oci8i — j8i7i 
 
 > 1 
 
 {fXi + 52)(q;i5i — /3i7i) — §(q:i + 5i)(ai52 + o;25i — /SoTi — /3i72) = 0. 
 The condition for hyperbolic motion is then that, in the transformation 
 
 yw -\- 5 ' 
 [hicc + 5)]2 
 
 ab — fiy 
 
 > 1. 
 
 For the case in which the motion is along points at infinity, the form of 
 the transformation 
 
 ^ (1 + ^y)w + («T - fiy^ - y) 
 
 is used. If this transformation is identical with the transformation 
 
 ^ y'w-{-8" 
 then 
 
 8(1 + /St) = «', 
 
 diay - ^7' - 7) = ^', 
 
 8^ = 7', 
 
 8(a - ^7) = 8', 
 and 
 
 a - 1 a'8' - ^'y' 
 
 (a + 1)^ ~ («' + 8'f • 
 If a is of the form 1 + ja^, then 
 
 • _ ^'^' - ^'y' 
 
 For the motion along points at infinity the condition is, dropping accents 
 and thereby taking the transformation to be 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 35 
 
 yw -j- 5' 
 that 
 
 a8 — ^y 
 
 contains j. 
 
 The general transformation 
 
 aw -\- 
 
 W = 
 
 yw + 5 
 
 contains three essential constants of the form 771 -]r j'fh- There are then 00" 
 such real transformations. 
 
 In the general transformation, four constants determine the asymptotes, 
 one constant determines the system of curves (C), and one constant deter- 
 mines the extent of the motion on the system. 
 
 One condition determines hyperbolic motion. Four constants determine 
 the system of curves and one constant determines the extent of the motion. 
 
 Two conditions are required for parabolic motion. Two constants deter- 
 mine the common element of the system of second involutes of circles, one 
 constant determines the distance to the common point of tangency, and one 
 constant determines the extent of the motion. 
 
 All linear transformations, being obtainable from combinations of the 
 transformations W = w -\- ^, W = aw, and W = Ifw, of course possess the 
 equilong property. 
 
 {e) The Simplest Invariant of a Regular Analytic Curve under the General 
 
 Linear Transformation. 
 
 The invariant of lowest order for the general linear transformation will 
 next be derived. All the linear transformations which leave the line (0, 0) 
 unchanged are represented by the equation 
 
 W = ; — : = aw(l + yw)~^. 
 
 yw + I 
 
 Expanding the right-hand member, the transformations may be written 
 U= Eai(-7i)'"w'"+S 
 
 00 
 
 F = Z) Hn + l)ai{- 7i)" - (nai(- T^^'^Ta - «2(- yi)'')u]u''. 
 
 n=0 
 
 If such a transformation turn the analytic curve 
 
 00 
 V = ^ hkU^ 
 
36 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 into the analytic curve 
 
 k=l 
 
 then the result obtained by eliminating TJ, V, and v among the four equations 
 is an identity with respect to u or 
 
 Z I E h^in + l)ai(- 7i)« - (p - l)ai(- 71)^^2 + a^{- 7i)^' 
 
 p=l I n=0 
 
 - £ 5^1' /' (- y.f'^ . • • (- Ti)*^ 1 u^ = 0, 
 
 a=l XqI' ' 'Xtl J 
 
 a?o. • • • , art being positive integers or zero satisfying the relations 
 
 Xi + 2«2 + " ' tXt = p — Sy 
 
 Xq-\- Xi-\- " ' Xt = s. 
 
 If the coefficients of the first five powers of u be each equated to zero, the 
 constants (oci, a^, 71, 72) of the transformation may be ehminated with the 
 result 
 
 W-hh B^-BzB,' 
 
 The left hand member of this equation is therefore invariant under the trans- 
 
 00 
 
 formations. In terms of the derivatives of the curve v = ^ hkUk at the 
 
 *=i 
 line (0, 0), this invariant has the form 
 
 3 
 
 J5 = 
 
 
 /^Y_ /^\/d^y 
 
 The transformations IF = «) + j3 and W = 1/w are not included in the 
 transformations W = aw{l + 7^)"^ For the transformations W = w -\r ^ 
 and W = 1/w, direct computation shows that J5 is invariant. 
 
 Hence the invariant of lowest order for the general linear transformation 
 W = (aM> + /3)(7W + 6)-i is 
 
 ^6 = 
 
 (dhV 
 \du^) 
 
 /^Y_ /^Wrf^y 
 
 A family of curves characterized by the property 
 
 Js = T (a constant) 
 
CONTRIBUTIONS TO EQUILONG GEOMETRY. 37 
 
 is invariant under the general linear transformation. By solving the dif- 
 ferential equation 
 
 \du') ^ 1 
 
 the 00 ^ curves of the family are found to be 
 
 v= — 
 2k 
 
 ;^[(..+ 6)2 + ^][log(i.+ 6-^AP^)-log(i^+64-^V-A:)] 
 
 a 
 a, h, c, d, and e being arbitrary constants. 
 
 + i§-^^ + ^[d+^]i. + e, 
 
38 CONTRIBUTIONS TO EQUILONG GEOMETRY. 
 
 VITA. 
 
 Paul Henry Linehan was born in Boston, Massachusetts, January 15, 
 1879. He attended the public schools of his native city and Harvard Uni- 
 versity, from which he received the degree of Bachelor of Arts in 1902. 
 Since February, 1903, he has been, successively, tutor and instructor in 
 mathematics in the College of the City of New York. 
 
L5- 
 
 UNIVERSITY OF CALIFORNIA LIBRARY