CURVES WITH A DIRECTRIX 
 
 DISSERTATION 
 
 Submitted to the Board of University Studies of the Johns Hopkins University 
 
 in conformity with the requirements for the deg;ree of 
 
 Doctor of Philosophy 
 
 BY 
 CLYDE SHEPHERD ATCHISON 
 J907 
 
 BALTIMORE, MD., V. S. A. 
 1908 
 

To Professor Morley, without whose helpful suggestions and constant 
 inspiration this paper would have been impossible, and to Dr. Cohen, Dr. 
 Hulbert, and Dr, Coble for their encouragement in his university course, the 
 author desires thus to express his thanks. 
 
 186885 
 
Digitized by the Internet Archive 
 
 in 2008 with funding from 
 
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/ OF THE 
 
 I UNIVERSITY 
 
 Curves ivith a Directrix. 
 
 Br Clyde S. Atchison. 
 
 Introductory. 
 
 It is a well-known fact of elementary geometry that the orthocentre of any 
 three lines of a parabola, which is of course uniquely determined by four lines, 
 lies on a line, called the directrix of the parabola. In his Ortliocentric Properties 
 of the Plane n-Line, Professor Morlet* has extended the meaning of the term 
 orthocentre, so that with every odd number of lines there is associated an ortho- 
 centre, and with every even number of lines a directrix. It is my purpose to 
 discuss the class of curves, of which the orthocentre of a certain number of lines 
 shall always lie on a line, the directrix of the curve in question. As the curves 
 to be considered are entirely rational, throughout the discussion, I shall employ 
 almost exclusively the method of vector analysis, worked out by Professor 
 MoRLEY, which is especially convenient for the purpose. The line equation of 
 a curve is here uniformly expressed by means of conjugate coordinatesf, while 
 the point equation is expressed by means of what is commonly called the map 
 equation ; that is, a point of the curve is expressed as a rational algebraic function 
 of a parameter t, which is limited to the unit circle, and is, therefore, always 
 equal in absolute value to unity. Throughout y and h will be considered as 
 conjugates of x and a respectively; that is, if X and Y are rectangular coordi- 
 nates of a point, then 
 
 x=zX -\-iY, 
 
 y = X-iY, 
 and similarly for a and h. { 
 
 * Transactions of the American Mathematical Society, Vol. IV (1903), pp. 1-13. 
 
 f F. Fbanklin: Some Applications of Circular Coordinates, ^mcWcan Journal of Mathematics, Vol. XII 
 (1890), p. 161; F. Moklby : On the Metric Geometry of the Plane n-Line, Transactions of the American Society, 
 Vol. I (1900), p. 97. 
 
 t H. A. Cosvbrsb: Annals of Mathematics, Ser. II, Vol. V (1904), pp. 106-109, where is given a fnlle 
 explanation of these methods. 
 
2 Curves with a Directrix. 
 
 Employing the notation used in the memoirs of Prof. Morley, a line a is 
 written in the form 
 
 where x^ is the reflection of the origin in the line, and — 1/<„ is the clinant. 
 
 The definition of the first orthocentre, as extended for 2n — 1 lines, is the 
 point of intersection of the In — 1 perpendiculars, let fall from the centres of the 
 2n — 1 A^"~"s, one of which is uniquely determined by every 2n — 2 lines, upon 
 the line left out in each case. By a* A^"~^ is meant a deltoid of order 2n — 2 
 and class 2n — 3, having 2n — 3 cusps, and containing the line at infinity as a 
 special {In — 4)-fold line, being tangent to it n — 2 times at / and n — 2 times at /. 
 Thus, in the case of three lines, the three A"s each of which is determined by 
 two lines of the three, are simply points, the vertices of the triangle of the three 
 lines, and the orthocentre is the intersection of the perpendiculars dropped from 
 the vertices upon the opposite sides. But in the case of five lines, every four 
 lines uniquely determine a A', and the perpendiculars dropped from the centres 
 of these five A^'s upon the line omitted in each case meet in a point, which is 
 the orthocentre of the five given lines. 
 
 For 2n — 1 lines, the first orthocentre* is of the form 
 
 i'n-i = «! — ^j ag + A^g ag — + (— )" ( + /S'„_2 a„_i — S^-i a,), 
 
 where Si stands for symmetric functions of the 2n — 1 <'s, and the a„ are the 
 so-called characteristic constants of the lines, and in the case of 2« — 1 lines 
 
 2n-l 
 
 x,tl 
 
 2n-l-a 
 
 "« - 2 {t, - t,) {t, -t,).... (<i -l2„ _7) * 
 
 From the results deduced in the Orthocentric Properties, the equation of the 
 directrix for any even number of lines, 2n, may readily be written : 
 
 — kjj 
 
 where Pn-i is a point of the form of the orthocentre of 2n — 1 lines, but con- 
 structed for a 2n line, and g'„_i is its conjugate. It is evident, therefore, that 
 the clinant of the directrix of 2n lines is 
 
 *n + l 
 
 * Orthocentric Properties of the Plane n-Line, p. 5. 
 
^.. °' 
 
 Curves with a Directrix. 3 
 
 § 1. Equation of the Curve. 
 
 Suppose any even number of lines are given, 2n. They have a fixed 
 
 directrix, whose clinant is ^^ii. Allow one of the 2n lines to move subject to 
 
 the condition that the orthocentre of every set of 2n — 1 lines, chosen from the 
 2n, shall always lie on this directrix. The problem is to find the curve 
 enveloped by the moving line. Equating the clinant of the directrix to a con- 
 stant turn Ilk, we have 
 
 whence 
 
 
 \V - '!"' 
 
 'Rii't- 
 
 «n 
 
 — ^fln + l = 0. 
 
 JDUt 
 
 
 2n 
 «n = 2 ? 
 
 
 ^1 h 
 
 {h-h){t^-h)-.--{h-t^,,y 
 
 and 
 
 Then 
 
 2n 
 
 «»+!- ^{t,-t,){t,-t,)....{t,-t,,) 
 
 2n 
 
 •Cj \.*\ fC 1 1 J 
 
 an l^a, + ^- 2u{t,-t,)(t,~t,)....(t,-t,,)-^- 
 
 Considering as the moving line the line whose reflection in the origin is x^, 
 and whose clinant is — l/<i, this last equation may be written in the form 
 
 for a?! and ^i are the only variables, aj and a^ are symbolic, and have meaning 
 only in combination. 
 
 Now the condition that the equation of the line in the form taken, 
 
 shall represent a line is that 
 
 Substituting from this for t^ in the above equation, we obtain 
 
 «r' y'i -kx^ yr' = {a,y, + a,x,y^'-', (1) 
 
 which is the pedal of the curve enveloped by the moving line. 
 
4 Curves with a Directrix. 
 
 The equation of a line in circular coordinates with clinant — Ijt is 
 
 where Xj is the reflection of the origin in the moving line, and y^ is its conjugate. 
 The equation of any line in conjugate coordinates may be written 
 
 where ^, iq and ^ may be considered as trilinear line coordinates, having for 
 triangle of reference the triangle whose vertices consist of the 7 and J points on 
 the line at inflnity and the origin. In order for these two lines to be identical, 
 
 c 1 
 
 Dividing equation (l) through by a;f"~^yf"~^, and substituting — - iox — and 
 
 — ^ for — , the equation of the envelope of the moving line is obtained in 
 the form 
 
 This is a curve of class in — 2, and since the equation has no terms in ^ higher 
 than the first, contains the line at infinity as a (2n — 3)-fold line. The 2n — 3 
 points of tangency, obtained by equating 
 
 _^n-l^n-2^;5.p-2jjn-l 
 
 to zero, are on the line at infinity in the directions ^ = n — 2 times, jy = 
 n — 2 times, and j^z^hvi once, where Ijh is the clinant of the directrix. The 
 curve is therefore a special and specially placed (2n — 3)-fold parabola, being 
 tangent to the line at infinity n — 2 times at /, n — 2 times at /, and once in the 
 direction given by ^ = ^ jy. 
 
 Moreover, since the line at infinity enters as a (2n — 3)-fold line, and the 
 equation is of class 2n — 2, the curve is a special Jonquieres curve in lines. 
 Giving to this curve the name ^^"~^, we may state generally : 
 
 Theorem l. Given 2n lines, one of which is allowed to move subject to the 
 condition that the orthocentre of every 2 n — 1 lines chosen from the 2 n, shall always 
 lie on the directrix, the resulting curve is a K^^~^. 
 
Curves with a Directrix. 5 
 
 The simplest K^^"^, obtained when n = 2, is evidently 
 
 which is the equation of a parabola, tangent to the line at infinity in the direction 
 given by ^ =ky;. Now, a parabola, which has for the clinant of its directrix 1/k, 
 must have — 1/k for the clinant of the direction to the point where it is tangent 
 to the line at infinity, since this point must lie in a perpendicular direction. 
 That is, in the case of the parabola K^, the direction indicated by ^ = ky! is the 
 direction whose clinant is — 1/k. But since all curves K^"~^, including the 
 parabola, are tangent to the line at infinity in the direction given by ^^ky;, 
 this means that they are all tangent to the line at infinity at the point, the clinant of 
 the direction to which is — 1/k. 
 
 If the original 2n lines are chosen so that 1/k is equal to — 1, the 
 directrix is perpendicular to the axis of reals, and the direction indicated by 
 ^ = ky!, in which all curves K^'^-^ are tangent to the line at infinity, is the 
 direction of the axis of reals. Hereafter, every ^^""^, which is tangent to the 
 line at infinity n — 2 times at /, n — 2 times at J, and once in the direction of the 
 axis of reals, mil he called a Kl'^~^. 
 
 For n = 3 and ^= — 1, we have the general Jonquieres curve Kl, containing 
 the line at infinity as its triple line. Thought of as a triple parabola, the curve 
 is specially placed, being tangent to the line at infinity at /, /, and in the direc- 
 tion of the axis of reals. 
 
 Orthocentre of 2n — 1 Lines of a K^^-^, 
 
 Any one of the 2n lines might be allowed to move subject to the condition 
 that the orthocentre of every set of 2n — 1 lines, chosen from the 2n, should 
 always lie on the fixed directrix, and the resulting curve would be identically 
 the same, for the moving line must at different times coincide with each position 
 of the 2n — 1 fixed lines, since their orthocentre lies on the directrix. Inasmuch 
 as any one of the In lines might be allowed to vary for a time, then remain 
 fixed, and one of the others be allowed to vary, subject to the given conditions, 
 it is evident that the In — 1 lines are any 2n — 1 lines of the curve. Hence, 
 the theorem follows : 
 
 Theorem 2. The orthocentre of any 2n — 1 lines of a K^^~^ must always lie 
 on the directrix. 
 
 Thus, in the case of the K^, where n = 3 (that is, in the case of the 
 Jonquieres curve of the fourth class, having the line at infinity as a triple line), 
 the orthocentre of any five lines of the curve must always lie on its directrix. 
 
6 Curves with a Directrix. 
 
 §2. Some Theorems Concerning the Orthocentre of In — 1 Lines. 
 In the Orthocentric Properties, already mentioned, it was proved that the 
 
 cr 
 
 first orthocentre of 2n — 1 lines is of the form "~^ , when the lines are taken 
 
 as tangents of a A^"~^ By ^„_i is meant the sum of the products of the 
 In — 1 <'s, taken n — 1 at a time, where ijt^ is the clinant of the line a, and 
 similarly for S^^n-i- ^or the work of this section it is more convenient to take 
 the equation of the A^"~^ in the form 
 
 where the clinant of any line a is now t^ . Consequently, the orthocentre of 
 2n — 1 lines tangent to a A^"~^ is now Sn. 
 
 Considering three lines tangent to a A^, which may be any three lines in 
 the plane, since a A^ can be taken tangent to any four lines, their orthocentre 
 is Sz, where S stands for symmetric functions of the three t^s, which are the 
 respective clinants of the three lines. This may be written 
 
 where S now stands for symmetric functions of two Vs. If /Si vanishes; i. e., if 
 
 k + k= 0, 
 the orthocentre is evidently independent of the third line, and is 
 
 — f 1. 
 The condition on the two remaining lines, since the clinant of one is equal 
 to the negative of the other, is that they he perpendicular. This is nothing more 
 than the well-known case of elementary geometry, where the orthocentre of a 
 right-angled triangle is at the vertex of the right angle. 
 
 Considering next five lines tangent to a A^, which may be any five lines in 
 the plane, since a A^ can be taken tangent to any six lines, their orthocentre 
 is S^, where S stands for symmetric functions of the five ^'s, which are the 
 clinants of the five lines. This may be written 
 
 Sz + «6 S^ {S for 4 t's). 
 
 If S^ vanishes, the orthocentre is evidently independent of the fifth line, and is 
 
 /S'a {S for 4 ^'s). 
 
 This may be written 
 
 Si + hS^ {SiotZt'B). 
 
Curves witJi a Directrix. 7 
 
 If /Sj vanishes ; that is, if 
 
 which carries with it its conjugate relation 
 
 ti + h + h— 0, 
 the orthocentre is evidently independent of the fourth line, and is 
 
 ^ (^^for3<'s); 
 
 that is, ti <2 ^3 • 
 
 Since ^Si and S^ for three t's vanish, it follows that 
 
 and the orthocentre is 
 
 *!• 
 
 That is, the condition on the three remaining lines is that they form an equilateral 
 triangle. 
 
 Now the equation of any line tangent to a A" is 
 
 I — at -\- xt^ — y t^ -{- ht*' — t'^ = 0. 
 The three intersections of the three lines, which form an equilateral triangle, are: 
 
 x, = -^+ht\-\.t\, 
 n 
 
 h 
 x,^-^+h^tl + t\. 
 
 Their centroid is tl, which is the orthocentre. Therefore, the orthocentre 0/ any 
 Jive lines is independent of two of the lines, when the remaining three form an equi- 
 lateral triangle, and is the same as the centroid of the three paints of intersection of 
 the three lines. 
 
 Now consider seven lines tangent to a A''^, which may be any seven lines in 
 the plane, since a A' can be put tangent to any eight lines. Their orthocentre 
 is Si, where iS' stands for symmetric functions of seven t's. This may be written 
 
 Si + t^ S3 {S for 6 t'a). 
 
 If S3 vanishes, the orthocentre is independent of the seventh line, and is 
 
 /Si {Sror6t'B), 
 
 which may be written 
 
 Si+tA {S {or 5 t's). 
 
8 Curves with a Directrix. 
 
 If S3 vanishes, the orthocentre is independent of the sixth line, and is 
 
 /Si {S for 5 <'s). 
 
 This may be written 
 
 Si + <B ^3 {S for 4 t'&). 
 
 If aS's vanishes, the orthocentre is independent of the fifth line, and is 
 
 Si {S for 4 Vs) ; 
 
 that is, ti t^ <3 ti . 
 
 The vanishing of S^ , where S stands for symmetric functions of four t's, carries 
 with it its conjugate relation 
 
 Si = (S for 4 fa). 
 
 The vanishing of S3 for five t's may be expressed : 
 
 S3 + t,S,=:0 (Sfor^t's). 
 
 But in order for the orthocentre to be independent of the fifth line, it was proved 
 that S3 of four t's must vanish. Consequently, 
 
 Sz = {S for 4 t's). 
 
 From these three relations, 
 
 Si = 0, ^3 = 0, ^3 = (SfoT^t's), 
 
 it follows that the four remaining lines form an ordered 4-gon with equal 
 
 angles — , for the relation 
 
 4 • • 
 
 exists between their clinants. Consequently, the orthocentre reduces to 
 
 /4 
 
 The equation of any line tangent to a A' is 
 
 l — a2t + ait^--xt^ + yt^ — bit^ + b2t^ — t'' = 0. 
 The four points of intersection of the four lines which form an ordered 4-gon 
 with equal angles-^, are: 
 
 x,= ^ + J^lil^ILlL +b,itl-b,ii-l)t\-t\, 
 tl ti 
 
 :r, = -'!^ + ^li^=:i^^-b,itj-b,il + i)t\-t\, 
 
 0-3= 'b^^ + ^li-l+l) +b,itl-b,{-i+l)tl-tt, 
 
 t\ t^ 
 
 a;i , ai{i - 
 
 7f + t, 
 
 x, = -^^+ "^(\+^) -b,iti-b,{-l-i)tl-t\. 
 
Curves xjoiih a Directrix. 
 
 9 
 
 Their centroid is — 1\, which is the orthocentre. Therefore, the orihocentre of 
 any seven lines is independent of three of the lines, when the four remaining lines 
 
 form an ordered 4-gon with equal angles — , and is the centroid of the four points at 
 
 n 
 
 which are the angles — 
 
 7t 
 
 But in addition to the four - points of intersection of the four lines, there 
 
 7t 
 
 are two points where the lines intersect at the angle — . The centroid of these 
 
 two points is also the same as the orthocentre of the seven lines where three are 
 arbitrary, for these points are 
 
 _ «2 
 
 "ill C 1 , 
 
 x"= ^+bitl-t{, 
 '■1 
 
 and their centroid is — 1\, which is the orthocentre. 
 
 4-gon with equal angles -j-. 
 
 Passing to the case of nine lines tangent to a A', the orthocentre jS^, where 
 S stands for symmetric functions of nine t's, is independent of the sixth, seventh, 
 eighth and ninth lines, when the remaining Jive lines form an equiangular pentagon, 
 tlie equal angles being ~, and becomes tl. As in the other cases, this is easily 
 
10 Curves with a Directrix. 
 
 proved by calculation to be the centroid of the five points at which are the 
 angles — . 
 
 Now it is evident that the process we have been carrying on can be con- 
 tinued without limit, and that the orthocentre SnOf 2n — 1 lines tangent to a A^"~^ 
 is independent of n — 1 of the lines, and becomes ( — 1)""^^", when the remaining 
 n lines form an ordered polygon with equal angles — , and that the orthocentre is 
 
 always the centroid of the n - points of intersection of these n lines. 
 
 But it was observed in the case of four lines which formed an ordered 4-gon 
 with equal angles -, that while the centroid of the four - points was the ortho- 
 centre of seven lines, where three were arbitrary, the centroid of the two 
 ~ points was also the orthocentre of the seven lines. 
 
 Passing to the case of five lines which form an ordered pentagon with 
 equal angles -, there are five points where the intersecting lines make the 
 
 o 
 
 angle - , and five points where they make the angle -— . The centroids of both 
 
 these sets of points are the same, as is easily proved by taking the five lines as 
 tangents of a A^, and calculating the centroids of the two sets of points. 
 
 In the case of six lines which form an ordered hexagon with equal angles -r, 
 
 there are six angles -, six angles -— and three angles -— , and the centroid of 
 6 6 6 
 
 each set of points is the same, as can most easily be proved by taking the six 
 
 lines as tangents of a A', and calculating the centroid of each set of points. 
 
 The general scheme, for convenience printed at the end of this section, may 
 
 be written down in the case of n lines, which form an ordered polygon having 
 
 equal angles — . 
 
 The fact, proved in the particular cases, concerning the centroid of the sets 
 of points in the case of n lines which form an ordered polygon having equal 
 angles - , is evidently true in general, and may be verified in any particular 
 
 case, following the given method of proof. Hence, the theorem follows: 
 
 Theorem 3. The centroid of any one of the sets of points associated with n lines 
 
 which form an ordered polygon with equal angles - , is the centroid of all. 
 
 7% 
 
Curves with a Directrix. 11 
 
 Also, since the centroid of the n - points was proved to be the orthocentre 
 
 of 2n — 1 lines, the theorem follows: 
 
 Theorem 4. The orthocentre of 2n — 1 lines is independent of n — 1 of the lines 
 
 when the remaining n lines form an ordered polygon with equal angles - , and is the 
 centroid of any set of points associated with the n lines. 
 
 TABLE. 
 
 2 lines give 1 angle — = — . 
 
 3 « " 3 angles -^ — . ^ 
 
 4 « " 4 " -^ — , 2 angles -^— • 
 
 5 « " 5 " —I — , 5 " \" . 
 
 6 « " 6 « — ^, 6 « -^, 3 angles -^^- . 
 
 If tl n 7 <( ^ 7 (( '^^ 7 « _r.!! 
 
 • • S J ' ly > ' 7 ' 
 
 8 « « 8 " -^, 8 « ^-, 8 « ^^, 4 angles*^. 
 
 9 « « 9 « _|_, 9 '. -^, 9 « ^^, 9 " ^. 
 
 10 " « 10 " -^, 10 « -4J-' 1^ " -T^'^^ " r^'^^'^Slesf^. 
 
 2m " " 27» « ^i^^, 2m « -^^, 2m " -^^^ 
 
 2m ' 2m 27« 
 
 2 m angles ("^~ ■>'' ,« angles ^. 
 ^ 2m " 2m 
 
 (2m + l) « " (2m + l) " 3^, (2- + 1) " ^lll' (^'" + 1) " 2-^1' ' 
 
 (^- + 1) " TOT' ^^^'^+1^^"^'^^ 2^1- 
 
 § 3. General Discussion of the K^^. 
 
 In the first section, the equation of the JT^""^ was obtained in trilinear 
 complex line coordinates, and the orthocentre of any 2n — 1 of its lines was 
 proved to lie on the directrix of the curve. In the present section an equation 
 of the curve will be derived by another method, and a general discussion will 
 follow. Some special cases will also be considered. 
 
12 Curves ivith a Directrix. 
 
 Envelope of a A^"~^ Moving Under Parabolic Translation. 
 
 Consider the general case of a A^"~^ moving under parabolic translation; 
 that is, a translation in which one and therefore all the points of the A^"~^ 
 describe a parabola, although not the same parabola. A convenient form in which 
 to write the map equation of the general parabola for the present purpose is 
 
 X 
 
 r' 
 
 {r^ — ty 
 
 where t is a constant turn. This parabola is of unit size, but since the unit of 
 measurement may be taken arbitrarily, it may be considered as any parabola. 
 On dividing the map equation by its conjugate, and substituting the value of t 
 for which x becomes infinite, 1/t^ is found to be the clinant of the vector from 
 the origin to the point at infinity on the parabola. 
 
 The map equation of the general A^"~^ with its centre at the origin is 
 
 (-)"[na„_ir-^-(n-l)a„_2r-H- ••• + (-)"(•••• + 2a, ^ + ^J-....) 
 
 (n-2)&„_2 (n-l)&^i 1 
 
 •••• r=i ^ r J' 
 
 The condition for the envelope of the A^"~^ , moving with its centre on the 
 parabola 
 
 X 
 
 X ^ 
 
 T^ 
 
 {t^ — ty 
 
 while its orientation remains constant, is that 
 
 tDtf , . < A/ 
 
 — =r-^, = a real quantity = = 
 
 <iA./ ^ t,D^J 
 
 in the equation 
 
 X : 
 
 r^ 
 
 \r'-t 
 
 6i _ \ _ (« — 2)6„_a («— l)&„_i 1 
 
 ••')■■•• p-J "f- r J- 
 
 + 
 
 t^ ' 
 
 This condition is satisfied when 
 
 t = ti. 
 
 Accordingly the equation of the envelope of a A^"~^ moving under parabolic 
 
 translation, is 
 
 P^2+ (-r[««n-l<"-'-(«-l)«n-2<"-'+ • • • • + (-r(. • . . + 2a,< 
 
 , 6i_ \ («— 2)&„_a ■ (^— l)^n-i "| 
 
 T ^2 ) fn-1 T ^ J • 
 
 X 
 
Curves with a Directrix. 13 
 
 A vector from the origin to any point on the envelope can therefore be considered 
 as the sura of two vectors, one from the origin to the centre of the A^"~', and 
 the other from the centre of the A^"~\ which is a point of the parabola, to the 
 corresponding point of the A^"~^ 
 
 Since the A^"~^ contains the line at infinity n — 1 times as an isolated double 
 line, and since the parabola is tangent to it once in the direction 1/t^, the 
 envelope must contain it as a special (2n — l)-fold line, and may be thought of 
 as a special and specially placed {2n — l)-fold parabola, being tangent to the line 
 at infinity n — 1 times at /, n — 1 times at /, and once in the direction I/tt^^ 
 The envelope is of order 2 « -|- 2, as is seen on substituting from the map 
 equation in the equation of a line, and it is scarcely necessary to remark, that 
 on account of the singularities at / and J in the case of the A^"~^ and conse- 
 quently in the case of the envelope, at each point the n — 1 points of tangency 
 are equivalent to but n coincident intersections of the curve with the line 
 at infinity. 
 
 On multiplying the map equation of the envelope by t and subtracting the 
 conjugate equation, the relation 
 
 <»-y-^,+ (-r[|^-^^ +••••+(-)"(• •••+7^-«i^+----) 
 
 .... 4- a^2<"-i — 0^1^1 = 
 
 is obtained. This equation is self-conjugate, and on eliminating y by partial 
 differentiation with respect to t, gives again the map equation. This self- 
 conjugate relation is therefore the line equation of the envelope. From this 
 equation, the envelope is seen to be of class 2«, for, clearing of fractions, the 
 parameter t enters to the 2 nth power. Hence, it is perhaps better to think 
 of the envelope as a special Jonquieres curve of class 2n. When it has been 
 proved that the orthocentre of any 2 n + 1 of its lines lies on a directrix whose 
 clinant is — 1/1'^ it will further be known to be a K^'\ To introduce the proof 
 of this, we consider some special cases. 
 
 Directrices. 
 The equation of the directrix of the parabola 
 
 _ -r« 
 
14 Curves with a Directrix. 
 
 as the locus of two perpendicular lines, is evidently 
 
 a; = - 
 
 t 
 
 2' 
 
 The line equation of the envelope of a A' moving with its centre on this 
 parabola while its orientation remains constant, is 
 
 T^—t^ t 
 
 t^-y-:,^. + -^-a,t^ = o. 
 
 This represents a general Jonquieres curve of the fourth class, or, thought of 
 otherwise, a triple parabola tangent to the line at infinity at /, J, and in the 
 direction 1/t^ Now, from Theorem 4, §2, it is known that the orthocentre 
 of any five lines of the curve is independent of two of the lines, when the 
 remaining three form an equilateral triangle, and is the centroid of the three 
 
 — points of intersection of the three lines. The clinant of any line is \/t. The 
 
 o 
 
 clinants of any three lines, which form an equilateral triangle, are l/<, Ijat, 
 1/o^t. The three —points of intersection of these three lines are : 
 
 ^^= {r^-t)t'-c.t) +^V + ^^(^+")' 
 
 Their centroid is 
 
 i {«i2 + x^ + «3i) = :^vz-i3 = a:, say. 
 
 This then is the orthocentre. Allowing t^ to vary, x evidently traces out a line, 
 which is the locus of the orthocentres for every five lines of the curve, and which 
 is therefore the directrix of the curve. The clinant of the directrix is the clinant 
 of the direction of the point at infinity on it. The point at infinity is obtained 
 from the equation of the directrix when t^ = r^. Hence, substituting this value 
 for t^ in the expression 
 
 
 
 
 r" 
 
 
 
 X 
 
 = 
 
 
 ~t^ 
 
 -Tt^ 
 
 = 
 
 
Curves with a Directrix. 15 
 
 the clinant of the directrix is found to be 
 
 1 
 
 Now, the line equation of the envelope of the parabola and the A^ is 
 
 . t^-y- J^,-\l +h-<i.t' + a,t'=.0. 
 
 This represents a special Jonquiekes curve of class six, being tangent to the line 
 at infinity twice at I, twice at J, and once in the direction 1 /r^. The two points 
 of tangency at I, and likewise at /, are equivalent to but three coincident inter- 
 sections of the curve with the line at infinity. 
 
 Now, from Theorem 4, §2, it is known that the orthocentre ot any seven 
 lines of the curve is independent of three of the lines when the remaining four 
 form an ordered 4-gon with equal angles ^, and is the centroid of the two - points 
 
 of intersection of the four lines. The clinants of any four lines of the curve 
 which form an ordered 4-gon with equal angles ^ are Ijt, \/it, —lit, — Ijit. 
 
 One point — is given by the intersection of the two lines whose clinants are 
 
 \jt and — Ijt. The other — point is given by the intersection of the other two 
 lines, whose clinants are Ijit and — Ijit. These two points are 
 
 T^ &! .2 
 
 r^ , 6i , 2 
 
 Their centroid is 
 
 X 
 
 t" 
 
 v^ — t* 
 
 This, then, is the orthocentre. Allowing t* to vary, x evidently traces out a line, 
 which is the locus of the orthocentres of every seven lines of the curve. The 
 clinant of this directrix, found as before, is — l/r^ 
 
 Similarly, the locus of the orthocentre for any nine lines of the envelope 
 of the parabola and a A^ is 
 
 X = 
 
 T» 
 
 and its clinant is — 1/t^. 
 
16 Curves with a Directrix. 
 
 From these equations, it is evident that the orthocentre of any 2n + 1 lines 
 of the envelope of the parabola and a A^"~^ lies on a directrix, whose equation is 
 
 X = 
 
 ^8»i + l 
 
 2)1 + 2 /n + 1 ■ 
 
 ^2)1 + 2 _^ 
 
 Its clinant, calculated as above, is — 1/t^. Since the equations of the directrices 
 all give the same clinant — 1/t^, and since the absolute value of the reflection of 
 the origin in them, being | r | , is equal to unity, it is evident that they are all 
 the same. Hence : 
 
 Theorem 5. The envelope of a A^"~^ moving under parabolic translation is a 
 K^^, whose directrix is the same as the directrix of the parabola described by the 
 centre of the /^^^~^ . 
 
 Method of Plotting the K^"". 
 
 The line equation of the general parabola used above is 
 
 tr 
 tx—y— ^_^ = 0. 
 
 The line equation of the A^"~^ is 
 
 (_)« (a;f » _ ytn-l) - fn-l _ 1 _ (^, <2n^Z _ S^^_, <)+•••• 
 
 The line equation of the K^^ is 
 
 . . . + a„_2 1 
 
 n-l 
 
 a„_i p] = 0. 
 
 The tangents at corresponding points of these three curves are parallel, for the 
 clinant of each is the same, Ijt. Since it was proved that a vector from the 
 origin to any point on the envelope of the parabola and the A^"~^ is equal to 
 the sum of the vector from the origin to the centre of the A^"~\ and of the 
 vector from the centre of the A^"~^, which is a point on the parabola, to the 
 corresponding point on the A^"~\ it is evident that by drawing a tangent to the 
 ^n-i ^ff\yxc\x is parallel to the tangent to the parabola at the centre of the A^*^^, 
 a corresponding point of the A^"~^ is thus obtained, which is a point of the -£"*". 
 As the centre of the A^"~' takes all positions on the parabola, while its orien- 
 tation remains constant, all points of the ^^" are thus obtained. 
 
18 Cv/rves with a Directrix. 
 
 Consider now the more general question of the envelope of a A^"'~*, whose 
 orientation is constant, and whose centre is moving on a ^^". The condition for 
 
 the envelope is that 
 
 t D f 
 
 . r! J- = its conjugate 
 h ^ti J 
 in the equation 
 
 X 
 
 + (— rfma:., er' - (» - l) a;^, (,-' + •■•• 
 
 + (-)•>(....+ 2a.' >. + I -....)... - ('"-ft^--' + C"-^) ^•■- ] . 
 This condition is satisfied, as before, when 
 
 That is, the equation of the envelope is of the form 
 
 It is evident, therefore, that when m<^n, the equation reduces simply to 
 
 and is another K^, with the same directrix as the E^^ over which it was trans- 
 lated, for the equation of the directrix is independent of the constants of the 
 deltoidal term. 
 
 When m^n, the equation of the envelope reduces to 
 
 "f j_ A2m-1 
 
 {r^—tf 
 
 n. — ^ _1_ A2m— 1 
 
 and is evidently a K^^ with the same directrix as the iT^" over which it was 
 translated, since both curves can be obtained by trao slating the proper deltoid 
 over the same parabola, and, consequently, according to Theorem 5, have the 
 same directrix. Hence, we may state the theorem : 
 
 Theorem 6. The envelope of a A""'"'^ whose orientation is constant, and whose 
 centre moves on a K^", is another K^" if nK^n, and is a K^'" if my-n, and the 
 directrices of all curves generated thus are the same; namely, the directrix of the K^". 
 
Curves voith a Directrix. 19 
 
 It should be observed that any curve jK"2"+2 can be obtained by translating 
 a cycloid 
 
 (n+l)a„P+-^ 
 
 over a K^^. 
 
 Now, in the equation of the ^^", if l/r^ be taken equal to unity, the 
 directrix is perpendicular to the axis of reals, and the curve is tangent to the 
 line at infinity n — 1 times at /, n — 1 times at J, and once in the direction of the 
 axis of reals, and is then designated by jff'g", according to the notation of § 1. The 
 equation then takes the form 
 
 X = J~(f + (-)" ["«"-! *"'' - (»^ - 1) «»-2 «""' + • • • • 
 
 § 4. Caustics* 
 Any rational curve can be represented by a map equation 
 
 which of course carries with it its conjugate equation 
 
 y=m 
 
 and the general form of its caustic by reflection, where the incident rays are 
 parallel to the real axis, can easily be derived. Letting t" represent the unknown 
 clinant of the line tangent to the curve at the point 
 
 /(O, 
 
 the equation of this line can be written 
 
 x-f{t) = t'\jj-f{t)-], 
 
 which, for varying t, is the line equation of the curve. On dividing through by 
 t% and eliminating y by partial differentiation with respect to t, the equation 
 
 or 
 
 .^f^e>-'m+'nm 
 
 is obtained, which must be the map equation of the curve. But, by hypothesis, 
 
 x = f{i) 
 
 ♦Salmon: Higher Plane Curves. Heath: Oeometrical Optict, F. Morley: On the CauHie of the Epicycloid. 
 
20 Curves with a Directrix. 
 
 is the map equation of the curve. It follows, therefore, that 
 
 tnt)^ i^^'f{t) 
 
 % z ♦ 
 
 whence 
 
 ~m' 
 
 This then is the clinant of the tangent to the curve at any point given by t. 
 Since when the incident rays are parallel to the real axis, the reflected rays 
 make twice the angle with the axis of reals that the tangents to the curve at the 
 points of reflection make with the axis of reals, it is evident that the clinant of 
 any ray after reflection at the point given by t, is 
 
 K]' 
 
 -ft. 
 
 and its equation is 
 
 which, as t varies, is the line equation of the caustic of the given curve. Since 
 the condition for the cusps of the curve 
 
 obtained by equating or to zero, is 
 
 m = 0, 
 
 which of course carries with it its conjugate equation 
 
 /'(O = 0, 
 
 it is evident further that: 
 
 Theorem 7. The cusps of any rational curve lie on its caustic. 
 
 Now, consider the case of the K'^^\ The condition for cusps is of degree 
 2n + 2, showing that the K^^ has 2n + 2 cusps, although not necessarily all real. 
 The line equation of the caustic by reflection, formed as in the general case, is 
 
 X 
 
 - 1 A. [ <' ,1 (_Y\^^n-i (n— l)6n_a 
 
 - t^\y \{\ — tf^^ 'I t^-i - f.-2 ^ • • • • 
 
 + (»-l)a„_iP]}), 
 
Curves with a Directrix. 21 
 
 which may be written in the form 
 
 t'x-y = i-Y ^na„_, <»+» - (n - l) (a„_, + a„_i) T 
 
 + (« - 2) (a„_3 + a„_2) <"-! -.... + (— )» j . . . . + 2<» (aj + a,) 
 
 + ^"^ (V. + &„-i) - "^11^- a, t\ 
 Its map equation is 
 
 + (»-2H"-l) (a„_3 + a„_3) <"-3_. . . . + (_)» I . . . . + 3<(ai + a,) 
 
 (n— 1)(« — 2),, , , \,n(n—l), ~\ 
 
 showing that the caustic is a cycloidal curve. It is to be observed, that in 
 
 forming the equation of the caustic of the K^^, since the parabolic term, j- t^, 
 
 f 
 appeard on the left side of the equation, and its conjugate, /.. ...g , appeared on 
 
 the right multiplied by Ijf, these terms canceled each other. Hence, since 
 the ^" is made up only of a parabolic term and a deltoidal term of class 2n — 1, 
 it is evident that the caustic by reflection of the general A'^""^ with centre at the 
 origin is the same as the caustic by reflection of the Kl", and is the cycloidal curve on 
 which lie the 2n — 1 cusps of the A"'~^ and the 2n + 2 cusps of the A'§". 
 
 In general, given any self-conjugate equation, which is a polynomial in t, 
 thus: 
 
 G: ajr— as^-^+agr-"— . . . . +(—)*( a;^ <""*+! +»<"-* 
 
 -a^+a <"-'-'+ • ...)•••• +(-)"-*( b^^.f^' + yf-b.t^-^ 
 
 + ....)....+(-)"(• ••• + &3<'-&2< + ^) = 0, 
 where a and b are conjugates, it represents a cyclogen, the clinant of any one of 
 whose lines is ( — )"l/<"~^*. Considering the incident rays as coming from the 
 point at infinity on the axis of reals, the clinant of the line equation of the 
 caustic of any (7„ is therefore 1/^201-2*) q^ writing out a few caustic equations 
 
22 Curves vyith a Directrix. 
 
 for the less complicated cyclogens, it is readily observed, that the caustic equations 
 also always represent cyclogens, and that if y is the coefficient of t'' in the C„, 
 it is also the coefficient of t^ in the caustic. Further, considering the caustic as 
 a polynomial in t, it is observed that the middle term is always absent, and that 
 the class of the caustic is equal to the sum of the order of the C„ and of the order 
 of t in its clinant ; that is, the resulting caustic is of degree 2n — 2^ in t. Hence, 
 the general form of the caustic of the general (7„ may be written, thus: 
 
 aj«i!n-2*_a2«2"-2'=-i + a3<2"-2*-^- .... +(-)'(n-7^- |)(a;+a,+i) <'""'*+ • • ■ • 
 
 + 0<"-* + . . . . + {-f{n-h-^{y+^,^,)t'^ + . . . ■+^,t'-^,t + (i, = 0, 
 where a and j8 are conjugates. 
 
 §5. Discussion of the K\. 
 
 (a) Cusps. 
 The map equation of the K\ is 
 
 The condition for cusps, obtained by equating ^ to zero, was found to be 
 
 (j^.+''-^=0. (C) 
 
 which is a sextic in t, showing the -ff"* has six cusps, although not necessarily 
 all real. There are just as many real cusps as equation (C) has roots <{ which 
 are real turns. 
 
 Considering a and h as variables, and calling them x and y, equation (C) 
 may be written - 
 
 {1—tf ' t' 
 
 For varying t, this is the line equation of the curve whose map equation is 
 
 This curve, composed of two loops, is tangent to the line at infinity in the 
 
 direction of the real axis, has a double point at 
 
 X =■ 1/4, cuts the axis of reals again at the point 
 
 a; = — 1/16, and is a sextic in lines. Now a is any 
 
 point in the plane, and for any point a there are as 
 
 many f's, which are real turns, satisfying equation 
 
 (C), as there can be drawn real tangents to the curve 
 
 (A) from the point a. It is evident geometrically that if 
 
 "^~(i-0*' 
 
Curves with a Directrix. 23 
 
 where t may take any value from i to — i (that is, if a lies inside both loops), 
 no real tangents can be drawn to (A), and the K\ has no real cusps. But if 
 
 — — 1 
 
 a-x- (i_^)4, 
 
 where t may take any value from i to — i (that is, if a lies on the smaller loop), 
 two real coincident tangents can be drawn to (A), and the K^ has two real coin- 
 cident cusps. If ^ 
 
 «>-(]:=7/' 
 
 where t may take any value from i to — i ; and if at the same time 
 
 ''<-(T^' 
 whei'e t may take any value from 1 to i and from — i to 1 (that is, if a lies out- 
 side the smaller loop, but inside the larger), two real distinct tangents can be 
 drawn to (A), and the Kl has two distinct real cusps. If 
 
 _ _ 1 
 
 where t may take any value from 1 to i and from — i to 1 (that is, if a lies on 
 the larger loop), two real distinct tangents and two real coincident tangents can 
 be drawn to (A), and the K\ has two distinct real cusps and two coincident real 
 
 '"^P'- ^f a = a: =1/4, 
 
 that is, if a lies on the node of (A), two distinct pairs of coincident tangents can 
 
 be drawn to a, and the Kl has two distinct pairs of real coincident cusps. If 
 
 ">-(T:^' 
 
 where t may take all values (that is, if a lies outside both loops), four real 
 tangents can be drawn to a, and the K\ has four distinct real cusps. Consequently, 
 it is impossible for the K\^ to have more than four real cusps. 
 
 (b) Conic on the Cusps. 
 
 The line equation of the /fj may be written as a polynomial in t, thus: 
 at'^ — t^{x->ra) + t'^{x-\-y—\) — t{y-\-h)-\-h = Q. 
 The invariant g.;^ of this quartic in t is 
 
 ^l (x + a)(y-f fe) ) {x + y-iy ^ 
 
 whose vanishing expresses the condition that four Vs be self-apolar. Hence, 
 
 the clinants of the four tangents to the Kq from any point on the conic 
 
 , {x + a){y + h) {x + y--iy _ 
 ab ^ + ~ 
 
The Kq with four real cusps. 
 24 
 
Curves with a Directrix. 25 
 
 are self-apolar. But from a cusp of the K\ three tangents coincide, and the four 
 clinants are self-apolar regardless of the position of the fourth tangent. There- 
 fore, this conic passes through the six cusps of the K\. 
 
 (c) Nodes. 
 From consideration of PLiJCKER's equations, we know that the Kl has four 
 nodes. Since the equation 
 
 '—s'-r^, + '-»''=" 
 
 is of class four, there are in general four tangents to the K^ from any point of 
 the plane. From a node, however, two pairs of tangents coincide, and so there 
 are only two distinct tangents. Hence, if the equation of the Kl be written as 
 a polynomial in t, thus : 
 
 ^'-^'(^+l) + l'^'' + 2/-l)-~(2^ + &) + ^ = 0, (B) 
 
 having for roots <i (where i = 1, 2, 3, 4), then at a node, 
 
 <i = <3 , and t^^ ti. 
 Equating symmetric functions of these roots to the coefficients of (B), we have 
 
 20^1=1 + ^, (1) 
 
 a? + 2(r, = ^+f^, (2) 
 
 2«ri(T, = 2^, (3) 
 
 .l = \, (4) 
 
 where 0^ = 1^ -\- 1^, and az = tit^. Eliminating (Ti and a.^ by means of equations 
 1, 3 and 4, the equation 
 
 ^=\ 
 
 X + a 
 
 is obtained, which evidently represents two perpendicular lines, passing through 
 the point — a, on which the four nodes lie ; that is, two lines whose intersections 
 with the line at infinity are apolar with / and /. But there are two other pairs 
 of lines, which are imaginary, on which the four nodes lie. Projectively, the 
 points /, J and K on the line at infinity, where K is the point in the direction 
 of the axis of reals, behave in a symmetrical manner. Therefore, it is evident 
 that the intersections of one of these pairs of lines with the line at infinity are apolar 
 with J and K, and that the intersections of the other pair are apolar with K and I. 
 
 JoHsa Hopkins Univbrsitt, January, 1907. 
 
Vita.. 
 
 Clyde Shepherd Atchison was born June 28, 1882, at Carnegie, Penna. He 
 attended the public schools, and graduated from Carnegie High School in 1899. 
 In September of the same year he entered Westminster College, Penna., from 
 which he graduated with the degree of A. B. in June 1903. For the past four 
 years, he has been a graduate student in Mathematics, Physics, and Psychology 
 in the Johns Hopkins University. 
 
 January 3, 1907. 
 
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