t 
 
 A GENERAL CYCLIDE 
 
 WITH SPECIAL REFERENCE TO 
 
 THE QUINTIC CYCLIDE 
 
 By 
 
 HARVEY PIERSON PETTIT 
 
 A.B. Kalamazoo College 1914 
 A.M. University of Kentucky 1919 
 
 THESIS 
 
 SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS 
 FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS 
 IN THE GRADUATE SCHOOL OF THE UNIVERSITY 
 OF ILLINOIS, 1922 
 
 URBANA, ILLINOIS 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 UNIVERSITY OF ILLINOIS 
 
 THE GRADUATE SCHOOL 
 
 _191 
 
 i hereby recommend that the thesis prepared under my 
 
 SUPERVISION BY. '-ettit 
 
 ENTITLED A GENE RAL TO LI TE! with special reference to 
 
 TIIS QU I NT 1 0 CYCm?-. 
 
 BE ACCEPTED AS FULFILLING THIS PART OF THE REQUIREMENTS POR 
 
 Recommendation concurred in* 
 
 Required for doctor’s degree but not for master’s 
 

 
 
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 . ■ ■ " w . 
 
 
 
 
 
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 ■ I 
 
 
 ' • 
 
 
 
 
 
 
 
 
 
 
 
TABLE OF CONTENTS. 
 
 I. Introduction 
 
 II. Tbs Quintic Cyclide 
 
 1. Tbs Double Tangent Cone 
 a. Direct Development 
 
 o. Mapping on a Ruled Cubic 
 
 2. Systems of Curves on the Quintic Cyclide 
 
 a. Curves of order 6,8. 
 
 b. Other Systems of Curves. 
 
 3. Minimal Lines on the Quintic Cyclide. 
 
 4 . Circles on the Quintic. 
 
 5. Special Quintic Cyc Tides, 
 a . First Special D a s e . 
 
 b , Inverse o f Ruled 0 u o i o . 
 
 c , Quin t i c fi i t. h 7 r t e x i r. i. . . . 7 a.i ical Flans. 
 
 III. General Cyclide 
 
 1. Projective Description 
 
 2. Systems of Curves on the General Cyclide 
 
 3. Locus of the Centers of the Generating Circles 
 
 4. Minimal Lines on the General Cyclide. 
 
 5. Further Generalized Cyclides. 
 
 15 
 
 18 
 
 20 
 
 2 a 
 
 pp 
 
 37 
 

 
 
 
 
 ' 
 
 ■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
I. Introduction. 
 
 The subject of the c ye 1 ides has been more or Is 
 ly trefted bv nearly all writers on the thee 
 has been treated from various standpoints ar 
 these surfaces have been fully discussed. The particular 
 surfaces which concern this pa per have not, feowever, received any 
 an: cunt of attention. 
 
 or 
 
 less e x h a 
 
 u s t i v e 
 
 of 
 
 surfaces . 
 
 It 
 
 t he 
 
 p r o p e r t i e 
 
 s of 
 
 In his c c u r a : 
 
 ur faces m tne year 
 
 .i e c l u r e s 
 
 rr. .~r s~. — r\ ca 
 
 • 1 . It I ' ‘ . 
 
 n c ^ »• "if 
 
 - iiSvi. ,y 
 
 A i as o r a i 
 
 :• la -> „ , r ret 0 3 S 0 r 1 ICC h CO 1 D f, 3 1 0 U I- Z h 3 
 
 projective description of the eyclides, indicating the extension 
 to tbs coin tie eyelids, of which tbs ordinary quartic eyelids 
 
 / j \ 
 
 forms a special esse.'- - ' 
 
 f fee quin tic eyelids, generated by a pencil of spheres and a 
 protectively related set of tangent planes to a cone, forms a 
 special case of a type of quintics discussed by ClsDsch^ 2 '. 
 
 St urm* 3 ^ , and Noether ^ .also considered the same type of ouintic 
 although they offered little except a recapitulation of the 
 results announced by Olebsch. 
 
 In the present paper it is proposed to consider t he 
 properties of the ouintic eyelids, its tangent cones, some 
 important curves lying on the surface, and some special cases of 
 the ouintic. Then the discussion is to be carried over, in some 
 part, to a mere general eyelids. 
 
 Cl) Tohoku Mathematical Journal. 7ol. 19, Nos. 1,2.:! ay 1 0 a 1 
 ( 2 ) Die Abb ild un <’. einer C 1 a s s c von FlScher 5 0 . , A b h a n d 1 u n r der 
 
 KBni.cf lichen Oesellschaft der riissenschaften a v. 06 ttin^en, V'ol.15. 
 ( S ) Wathemat ische ft ns a 1 o r V c 1 . 4 • 
 
 ( 4 ) Hatheiatische An r, alen V o 1 . 3 . 
 
Digitized by the Internet Archive 
 
 in 2016 
 
 https://archive.org/details/generalcyclidewiOOpett 
 
2 
 
 IT. The Quintic jyclide. 
 
 § 1 „ The Double I a n g s n t Cone. 
 
 The quintic eyelids is represented by the equation 
 ( i X »,D 2 -i- opn j. v r 2 - n 
 
 where a, 8, y are linear expressions in x.y.z and F and G are 
 
 "if 
 
 X n = n 
 
 A C. V> 
 
 the form R 2 -A and R 2 -8 respectively, where R 2 = x 2 + ,y 2 + z 2 and A, 8 
 are linear in x, y,z. The- surface is pre.j eetivs iy described by the 
 cone and pe no i 1 of spheres 
 ( 2 ) 
 
 1 he eq nation of the cone proper is 
 ( 8 ) ?. 2 - 4 e y = C 
 
 and the vertex is the point a=8=y=0 which lies on the surface 
 a s is evident from equation ( 1 ) , 
 
 Any line through the vertex of the cone meets the quintic 
 in four other points which lie on the generating circles in the 
 two planes tangent to the cone and intersecting in the line. As 
 the two planes approach coincidence the circles corns together 
 sc that a generator of the cone meets the quintic in two 
 double points. 
 
 Any two such planes are 
 a A 2 + 8 A +■ v = 0 
 a u 2 + 8 u + y = 0 
 
 ( 4 ) 
 
 From 
 
 these 
 
 two 
 
 equations we have 
 
 
 
 
 (5 ) 
 
 ; ; 
 
 y = 1 : - ( A + u ) : A u 
 
 
 
 1 — 1 
 
 hese values 
 
 are substituted in 
 
 squ a 
 
 t ion ( 1 ) the r e s u 1 1 i s 
 
 
 (6) 
 
 F 2 
 
 - A PQ - |i?Q + AnS 2 
 
 = (F 
 
 - AG. ) (P - uO) = 0 
 
 which shows that the four points lie by pairs on the generating 
 circles corresponding to the values A , u of the parameter. 
 
Then if the two generating planes coincide, i .e . if A=u 
 equation (c) reduces to 
 
 (?) (P - AO) 2 = 0 (i) 
 hence we have the 
 
 Theorem 1. The cone 3 2 - 4ocy = 0 is a double tan 
 Theorem 2. Every plane of the Generating cone cuts the 
 quint ic eyelids in a circle and a cubic which lies on a cubic 
 eye lide. 
 
 Consider the plans 
 
 ( 6 ) a A 2 + 3 A + y = 0 
 w i t h the s u r f a c e 
 
 ( 9 ) a P 2 + 8 F Q + y Q 2 — 0 
 
 Eliminating y between (5) and (9) we have 
 
 (10) o:P ? 
 
 p ppi ^ \ JA2 p \ p 2 ■— ( 
 
 A Q ) ( oc P + a A Q 
 
 The second factor in (1C) is of the form 
 (11 ) a 'F + D ’0 - 0 
 
 which is the equation • of a cubic cyclide. 
 
 Theorem 3. The curve of contact of the double tangent cons 
 with the quint ic cyclide is of order 5. v2 ^ 
 
 The equation of 
 
 the quirt ic eyelids 
 
 is 
 
 (12) o-.F 2 + 
 
 8PQ + yQ 2 = 0 
 
 
 while the equation c 
 
 f t h e double tangent 
 
 cone is 
 
 (13) 32-4 
 
 OC Y — 0 
 
 
 Multiply (22) by 4a 
 
 and rearrange terms. 
 
 T he res u 1 
 
 (14) ( 2 a F + 
 
 BO) 2 - (3 2 - 4 o: y ) C 2 
 
 ii 
 
 o 
 
 ( 1 ) The analytic treatment of theorem 1 follows the w c r If of 
 
 Clebsch. loc.oit. p p 9,10. 
 
 Theorem 3 due to Clebsch. The proof, however, is new. 
 
 v ~ / 
 
 • re. 
 
4 
 
 The curve cf contact is the intersection of tne double tangent 
 cone (13) with the cubic eyelids 
 (15) p ocP + 30. = C 
 
 This intersection is of order 6, but the line ot=B=0 factors out, 
 leaving a proper intersection of order 5. 
 
 Theorem 4. The proper single ton§eni cone with vertex 
 a=?=C is of order 6. ' 1 
 
 The- 
 
 intersection cf 
 
 the fi 
 
 rst oolar of the 
 
 oo i 
 
 nt a.=? = y-C 
 
 ith the 
 
 quin tic c.yclide 
 
 is of 
 
 order 30, reduced 
 
 b.y 
 
 3, since the 
 
 point lies on the puintic. Then the cone with this point as 
 vertex and tangent to the quint ids has this same curve as curve 
 of contact. But there breaks off from this cone, the cone through 
 the nodal curve , counted twice and the double tangent cone 
 
 counted 
 
 twice. This reduces 
 
 t he 
 
 ^ V U ^ <-> 
 
 y 2 s 
 
 I '•■> 
 
 •I- 
 
 tO 
 
 CO 
 
 II 
 
 2 and leaves 
 
 v» m rs ir» v* 
 
 C i. v y v : 
 
 s i. n g 1 s t a n g e n t c c n e 
 
 c £ 
 
 order 
 
 6 . A 
 
 direct 
 
 analytic proof 
 
 of this 
 
 theorem i s given by 
 
 Gleb 
 
 s c h . 
 
 
 
 
 Theorem 5. The c.urve of contact of the single i anient cone 
 with the Quiniio cuclide is of order 7. ( 2 ) 
 
 The defining curve of the single tangent cone is of order 
 6. Then any plane through the vertex meets this curve in six 
 points. But since one of the generating circles passes through 
 the vertex one line through the vertex is tangent to the 
 surface at that point. Bence the vertex itself must be counted, 
 as one point in the intersection with the curve of contact. 
 
 Thus the curve of. contact is of order 7. 
 
 (1) Clcbsch loc.cit.p 10. 
 
 ( 2 ) ibid. 
 
. 
 
 
Theorem 6. The locus of the centers of the generating cir 
 circles is a space curve of order 5. 
 
 In order to determine this locus, consider a line through 
 the center of one of the generating spheres and perpendicular 
 to the corresponding plane. The intersection of this peroendicul 
 lar with the corresponding plane of the tangent cone is the 
 center of the circular generator. 
 
 The equation of such a sphere may be written 
 (16) R 2 ( 1-A ) -2w -j x-2\y ? y-2>i/ R z-y/ 4 = 0 
 w here v % , w P , v/ a , v/ 4 are de f ined b y 
 
 V 1 h ) 4 f z ~ 1 A - 1 , 2 p P ~ >1 ? A 3 p , 3 W yj — a 3 A -• R , '•}/' 4. — H 4 A 4 
 
 rhe suoscra 
 
 1, 2,3,f indicating respectively the x,y,z and 
 
 constant coefficients . 
 
 The center of this sphere is the point 
 
 ?s; . Tfr ^ )!• _ 
 
 (20) 
 
 -A' 1-A 
 
 The cor responding plane is 
 
 ( 21 ) 
 
 r A 2 + ?X+ y = ® a.x + © 2 y+a> 3 z + <p 4 = 0 
 
 using the same subscript notation as above . It is evident that 
 
 the cd's are Quadratic in A and th; 
 
 The direction cosines c: 
 
 1 s linear in A . 
 
 -lane (21) are proportional 
 
 the coefficients tp*, Then the equation of the line in 
 
 iuesticn is 
 
 ( 1 -A ) x— y> , _ ( 1-A ) y-u ; o _ ( 1-A ) z—\y 
 
 ( 22 ) 
 
 Solving scuations (21) and (22) simultaneously for x,y,z 
 
 we find 
 
 cn cn 
 


 cx = 
 
 ( 1— A ) cp x cp 4 
 
 Cp 1 ( 'P ? W c> 
 
 * 
 
 
 ) 
 
 ~ Ti 
 
 (<pf 
 
 + cp? ) 
 
 ( 9 ' 
 
 ) = 
 
 ( 1 -A )cp p cp 4 
 
 + <P ? ( cp ! v{/ , 
 
 T 
 
 PpWp 
 
 ) 
 
 - <P? 
 
 ( m 2 
 
 \ y 3 
 
 + cp? ) 
 
 V 2 C . 
 
 DZ = 
 
 ( 1 -A )fp s <p 4 
 
 + 9 ? ( C P 1 w n 
 
 
 Cpp' 4'2 
 
 \ 
 
 j 
 
 ~ <Ps 
 
 (cp? 
 
 + ®|) 
 
 
 P t = 
 
 (l-A)(cp? + 
 
 qp | + <p | ) 
 
 
 
 
 
 
 
 and 
 
 since ti 
 
 1 e co 1 s are 
 
 quadratic 
 
 a 
 
 nd th 
 
 & 
 
 •j; 1 s 
 
 lin 
 
 e a r i n 
 
 expresses parametrically the equation of a space quin tic . 
 
 Theorem 7 . The normals from the centers of the spheres of 
 the generating vend l to the correspond inf fenera tint planes 
 form a ruled cubic. 
 
 Consider any plane 
 ( 24 ) ax + by + 2 + d = C 
 
 and the line 
 
 ( 3 5 ) 
 
 ( 1-A ) cppX - ( 1 — A ) ® a y 
 ( 1-A ) m 3 x - ( 1-A ) c x z 
 
 + X 2 w , 
 
 + D - \y . 
 
 Cl'. „ w , = 0 
 
 ( 26 ) 
 
 Eliminating z we have 
 
 ( 1 — A ) 0 / 2 x - (1-A)q; i y + y x y 2 - a ? w . = C 
 (l-A)<j) s x f (l-A)q>. 1 (ax+by+d) + <p i «|r 3 - sj 3 w - = 
 From which 
 
 ( 2 7 ) px = b ( cp ? \y 4 - cp A y s ) *. © , \ij 5 - m a p p - ( 1 - A ) d cr , 
 
 0 y = a 
 
 v ’.p p p P I J • Pp 
 
 "’p - (l-A)dp. 
 
 T bus any pi; 
 
 ( a vii - D cp 
 
 P V f 
 
 ) 
 
 
 
 
 r -j Q n r~ -p 
 
 t he 
 
 ruled 
 
 s ur f ace formed 
 
 by 
 
 t hese 
 
 r + h r» a g 
 
 and 
 
 ‘p op20 
 
 the ruled surf a 
 
 0 e 
 
 itself is 
 
 a cubic 
 
 The plane ( 24 ) cuts the places of the double tangent cone 
 in a class conic of which there are two lines through every point 
 of the cubic cut cut by the ruled surface of Theorem 7 . On the 
 other hand, any line of the class conic cuts the cuoic in three 
 qoints. Hence there is a three-two correspondence between the 
 ooints of the cubic and the lines of the conic. Then there are 
 
2. Systems of Curves on the Ouintic Cyclide. 
 a. Curves of order 6,9. 
 
 theorem 8 . There are - 1 7 a u int ic eye l ides in space. 
 
 A quin tic cyclide is determined by the pencil of spheres, 
 the double tangent cone and a pr objectivity between the spheres 
 of the pencil and the tangent planes of the cons. 
 
 The radical plane of the pencil of spheres can be chosen 
 in * s ways. Having chosen the radical plane, one can choose the 
 radical circle in ~ 3 ways and then the pencil of spheres is 
 completely detsrmi ned . 
 
 The vertex of the double tangent cone can oe chosen in 
 303 ways and the base of the cohe in « 5 ways. 
 
 Consider any three planes of the double tangent cone. 
 
 Each of these is cut by the spheres of the pencil in - 1 circles. 
 T he c ho ice of any one c i rcis in each of t he t hree planes 
 fixes the project i v i t y w h i c h vr. a y t here f ore be set up in * s w a y s . 
 Therefore t h e .c ui n tic c v c 1 i d e m a y o e d e t ermine d i n 
 
 cc 3 • cc 3 • oc 3 • ccS-cc3 = cclV n 3 V S • 
 
 Theorem 9. Any sphere cuts the quint io cyclide in the 
 sphero-circle at infinity and a pi. rite sphero-sextic which 
 lies of a. ruled c u b i c . 
 
 The equation of the quint ic cyclide may be written in the 
 
 form 
 
 (1) q:, r - a(R 2 -::) 2 + 8 (? *- V) (3 2 -S ) + v ( d 2 -8 ) 2 = C 
 and any sphere in the form 
 ( 2 ) R 2 — C = 0 
 
 Eliminating H z between the two equations we have 
 

 
 
 
 
 
 
 
 
 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
b . Mapping on a Ruled Cubic. 
 
 Consider the mapping of the quintie eyelid; 
 
 (1) aF 2 + 5PQ + yC 2 = 0 
 by means of the transformation 
 
 (2) x' = ^ 
 
 Y 
 
 y 
 
 y 
 
 f - 
 
 Then the surface ( I ) goes ever into the surface 
 
 ( 3 ) X ’ z ’ 2 + y ' z ' + 1 = C 
 
 a ruled cubic whose era tors are parallel to the x 1 y ’—plane 
 and tangent to the parabolic c y 1 i n d e r 
 
 (4) y ' 2 = 4x ’ 
 
 since t 
 
 h e y p r o j e c. t c n t h a 
 
 
 1 y '—plan 
 
 
 to 
 
 the 
 
 1 i n ° ^ 
 
 nvsl o 
 
 ping 
 
 the par 
 
 a b o 1 a i n w h i e h t h s 
 
 c 
 
 y 1 i n d e r 
 
 (i) 
 
 c u t 
 
 s t h 
 
 at p 1 a n 
 
 e . 
 
 
 T b 
 
 e transformation 
 
 r o 
 
 m t h e I 1 
 
 -spa 
 
 
 to t 
 
 re ~-so 
 
 a c e i 
 
 s a 
 
 1 -2 transformation. 1 r a t 
 
 I s 
 
 f q •:.} cj c 
 
 ngls 
 
 '0 0 
 
 int 
 
 0 L 0 ll v 
 
 ruled 
 
 cubic 
 
 there c 
 
 or respond two poin 
 
 is 
 
 of the 
 
 q u i n 
 
 tic 
 
 -j ' o 
 a _J or 
 
 c h l 5 -s n s 
 
 r a t o r 
 
 of 
 
 the cyl 
 
 i n d e r ( 4 ) is • 
 
 nt 
 
 to the 
 
 0 U b 1 
 
 c , 
 
 Then 
 
 ~ p 0/ 
 
 enera 
 
 tor 
 
 of the 
 
 cone 
 
 
 
 
 
 
 
 
 
 (5 i 
 
 22 - 4 a. y - C 
 
 
 
 
 
 
 
 
 
 w h i c h c 
 
 orresponds to (4) 
 
 is 
 
 d o u b 1 y 
 
 tang 
 
 ent 
 
 t o 
 
 the q ui 
 
 d t i c . 
 
 A nd 
 
 each circular generator of the quintie, which corresponds to a 
 generator of the ruled cubic, is tangent to the cons (5) ir two 
 
 points . 
 
 Other properties of the quintie cyclide could be deduced f r 
 from this mapping. It has seemed more advisable, however, to 
 proceed by direct methods. 
 
( ■< ) 
 
 ■ , - 
 
 (C— A ) (0-8 ) t( J-5 - 
 
 which is a ruled cubic genera tea by 
 
 (4) 
 
 V 
 
 _ ‘ ■ _ , 
 
 V ~ H ) 
 
 and which has for a double line 
 
 (5) C - k = 0 - B = 0 
 
 The ore or: 20 . Any sphere cuts the quint ic eyelids in a so hero 
 sex tic having two double points and which can be projected, into 
 a plane sextic of fenus 2. 
 
 By checrea 7 any sphere cuts the quin tic eyelids in a 
 spbero-sext ic which lies on a ruled cubic. But since the ruled 
 cubic has a double lire ns curve of intersection has two 
 d o u d 1 s noi r t. ? r . a pa r r - ■ o ■ • -• ■ p - ;r - •-» i p i. ? o r r> = > ■ r r- r • - p 
 
 s 7 3 r V D 0 1 T: ' 
 
 
 i 
 
 the ruled cubic which necessarily t 
 
 en era tor s 
 
 s sphere in 00111x0 
 
 X r* ! 
 
 il v 
 
 urve of intersection, 'then if the curve 09 pro.] 
 
 
 on 
 
 any plane from a point of the double line of the ruled cubic tbs 
 result is a plane sextic with a multiple point of order 4 and 
 two ordinary double points. But the multiple point of order 4 
 counts for 6 double points and then the sextic has 8 double 
 ooints and hence is o. genus 2 . 
 
 The double line of the ruled cubic lies in the radical 
 plane of the pencil of generating spheres and hence the double 
 points of the spbsro-sextic lie on the nodal circle of the ouint 
 
 ic . 
 
 Equation (1) can be rearranged in the form 
 ( 6 ) co * = (a+ 3 + y ) 3 4 — ( 2 a A + 6 ( A + 3 ) + 2 y B ) R 2 + a A 2 + 3 .A B + v 3 2 = 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
11 
 
 and we then have the corresponding form for the ruled cubic 
 
 (7) R s = ( a+o + y )C 2 -.[ 2aA + B(A + 3) + ?y3] C + a.4 2 + f3AB + yB 2 = 0 
 Subtracting (?) from (6) 
 
 (8) <p s -R s = ( a + 6 + y ) ( R 4 ~C 2 ) - [ 2<xA + 8 ( A+8 ) +2 y3] (R 2 -0 ) 
 
 = ( 1 $ -0 ) [ ( a + ? + y ) ( d 2 +C )-[ .2 a A + ?• ( A + 3 ) + 2 y 3 ] ] 
 
 = (R2-a)f>, 
 
 where is a cubic c.vclide 
 
 (9) a (R 2 — 3 A+C ) + ? ( 3 2 —a— 3+0 ) + y ( 3 2 — 38+0 ) = 0 
 generated o y 
 
 , , (a*K) + A(| + v) = C 
 
 ( 10 ) 
 
 (R 2 - 
 
 ^ A 
 
 ) _ A (R 2 - 23 + 0) = 0 
 
 and which cuts the quintic eyelids in the line 
 
 (11) A " 3 = ° 
 
 <x + 3 + y = 0 
 
 From equation (8) we have 
 
 (12) err, - R ~ + (R 2 - 0)(p a 
 
 That is, the quintic eyelids may be generated by the pencil 
 c u o i c s 
 
 (13) R .. + A «p R ~ 0 
 
 and the pencil of concentric spheres 
 
 (14) R 2 — 0 - A - 0 
 
 8 u t t fe is ns nc i 1 of co n c e n t r i c s o h s r e s c a n be chosen in ®= 3 tv ays . 
 And since the choice of 0 fixes R, and we have the 
 
 Theorsfli 11. f k e 
 
 quintic cyclide 
 
 may be generate 
 
 d in a Iri p l y 
 
 i n f i n it e n u in bsr o f 
 
 ways by a pencil 
 
 of cub ics ana a. 
 
 p r o J e c i i u e l y 
 
 related pencil of c 
 
 oncentric spheres 
 
 • 
 
 
 Theorem 12. The 
 
 whole system of 
 
 sphero-sext ics 
 
 l a in g o n 
 
 ruled cub ics can be 
 
 obtained as the 
 
 intersection of 
 
 the a u i n tic 
 
12 
 
 eyelid es and spheres . 
 
 There are, according to Theorem 8 == 17 quin tic eye Tides in 
 space. There are in space 0=4 spheres. Then we should expect 
 *=- 21 sphero-sext ics as the intersections of the spheres with 
 the eye tides. 
 
 There are == 1 3 ruled eubics in space. A r u 1 a d cud i c is 
 determined by two lines in space and a [ 2* 1J -correspondence 
 between their points. The two lines can be determined in °= 8 
 ways and the [2*1] -correspondence can be as ablished in <* s 
 w a y s , making ocl3 w a y s c f c b c o s i n g t r s ruled c u Die. 
 
 Bence there are actually only ocl7 sphere— sex tics of the 
 required type. 
 
 The apparent contradiction disappears when it is seen that 
 there are quint ic eye tides through any such sphero-sext ic . 
 
 lonsider the spher< 
 
 tic with two double points, r be 
 
 lie on the nodal circle of the quin tic c.yclide and on the nodal 
 line of the ruled cubic. The ruled cubic is unique because its 
 double line is fixed and all its generators are determined. 
 
 The sphere is unique because an infinite number of its non— 
 ccplanar points are fixed. 
 
 The ruled cubic may be written in the form 
 (15) <x (.4-0) 2 + B'(A-O) (B— 0) + y(B-C) 2 = 0 
 With A , B , 0 , o: , ? , y fixed, the ruled cubic and the cubic eyelids 
 and hence <p r are determined. The question must then be consider 
 ed as to the number of ways in which R, can be represented in 
 the form 
 
 ( 16 ) 
 
 ( A 1 -0 ) 2 + 3 , (A , -C)(B. , -C) -i- y'(B'-O) 
 
 2 = 0 
 
13 
 
 Since the 
 
 nodal line of R 
 
 3 is 
 
 f i x e d 
 
 we 
 
 may find A' ana B ! in 
 
 following 
 
 w a y . 
 
 
 
 
 
 (17) 
 
 A ' - C = A x ( A 
 
 - 0) 
 
 + u a ce 
 
 - 
 
 C) 
 
 
 B ' - C - A p ( A 
 
 - ^ ; 
 
 4 u ? ( B 
 
 - 
 
 C) 
 
 Substituting in equation 
 
 (16) 
 
 w e n a v 
 
 rrv 
 
 
 (18) 
 
 or'[A a (A-0) + «4(3 
 
 -C)] 
 
 2 + 3 ' [ A 
 
 (A- 
 
 -Ou,(B-0)] [X.(A-C) + Mp 
 
 + y 1 [ A ? ( A-0 ) + !i P ( 8-0 ) ] 2 - 0 
 
 Collecting in posers of (A-C) and (3-C) and comparing with ( 
 ■?i g find 
 
 (18) 
 
 a 1 
 
 A ? + 3 ' A a A 2 + y ' A | = a 
 
 
 2 a 
 
 J A - [i i + 2 1 ( A, a ? + A ? u , ) + 
 
 
 a ! 
 
 U f + 3 ' JJ 1 p j> -r V ' f.i | = Y 
 
 S o 1 v i n g t h 
 
 5S3 
 
 3ous t i. o n s for a 1 , 3 ' . y 1 
 
 
 
 a u \ — BArpj + y A f 
 
 ( 2 J ) 
 
 a ' 
 
 ( A , U p - A p M a ) 2 
 
 
 ,q i 
 
 -2aU a Up + ;3 ( Aifip + ApUi )• 
 
 
 
 ( A a u g - A , a i ) 2 
 
 
 T 
 
 o: u f - 8 A a u 1 + Y A ? 
 
 ( A 1 }' o ' ' p A j ) 
 
 
 which expresses o: ! 1 , y 1 explicitly in terms of X % , u x> u?. 
 
 Then since 
 
 (21) A 1 = A, (A-C) + ip(B-C) C 
 B' = A ? ( A -0 ) + (3-C) + 0 
 
 are con-homogeneous inA 1 , A ? , u a , ti P there are 4 essential para 
 and hence 3=4 ways of ?ir i ting the s am® ruled cu.b ic in the req 
 
 f or m . 
 
 But for 
 
 e v s r y 
 
 such c 
 
 h o i c s 
 
 of the R s there is a unique 
 
 q u i n t i 
 
 c c yc 1 i 
 
 de and 
 
 h a n c ° 
 
 t here 
 
 are 3=4 q uint ic eye 1 ides thro 
 
 every 
 
 sphere- 
 
 s e x t i c 
 
 of the 
 
 t y p e 
 
 under consideration. 
 
 Therefore the number of apparent sober o-sex tics cc?1 n us 
 be reduced to = 17 ,the total number of such curves. 
 
 t he 
 
 (3-0)] 
 
 1 ® ) 
 
 meters 
 
 u i r ed 
 
 ug h 
 
15 
 
 b. Other Systems of Curves. 
 
 Theorem 13. It is possible to secure infinite systems of 
 curves of any order equal to or treater than 2 on the quint ic 
 cycbide. 
 
 We have already seen that there exists a real straight line 
 on the quintic cyclide. We shall see in a later section that 
 there exist six pairs of minimal lines and one real pair. 
 
 Any generating sphere of the quintic cuts the quintic in the 
 nodal circle and a circular generator, fence there exists on the 
 cyclide an infinite system of circles. We shall also see that 
 there exists a finite number of circles net belonging to this 
 
 s y s o e m . 
 
 Any plane of the generating p 
 in a circle and a cubic which lies 
 exists an infinite system of plane 
 
 If a sphere be passed through 
 the remainder of the intersection 
 q u ar t i c whose degeneracy gives r i s 
 set of circles on the surface. If 
 line of the quintic tbs remainder 
 clans quartic. 
 
 In general a plane intersects 
 
 lanes intersects the cyclide 
 on a cubic cyclide. Be nee t her 
 cubic s on the quintic. 
 one of the generating circles 
 with the quintic is a sphere— 
 e to an interesting finite 
 a plane be passed through a 
 of the intersection will be a 
 
 the quintic in a curve of 
 
 order 5, 
 
 We have already noted an important class of s ext ics lying 
 on the quintic. A second class of sextics may be obtained as 
 
 follows . 
 
 A cubic cyclide intersects the quintic in the sphero-circle 
 
1 6 
 
 and a finite curve of order 11. If the cubic be mads to pass tb 
 through the nodal circle the curve of intersection, proper, is 
 
 cubic be made to pass 
 
 reduced to order 
 
 I 
 
 < . 
 
 I f , i n 
 
 addition, t 
 
 through the line 
 
 
 
 
 h _ R 
 
 (03) 
 
 - 
 
 0 
 
 
 a + 9 
 
 + 
 
 V = 0 
 
 
 the curve of inte 
 
 r s 
 
 action 
 
 is reduced 
 
 Such a cubic 
 
 3 
 
 yelide 
 
 is 
 
 (24) (a + 
 
 o 
 
 E 
 
 + Y - 1 
 
 ) d o - r\ 
 
 } u r bn — V./ 
 
 Consider its intersection with the quintic eyelids 
 
 (25) o: F 2 + 8PQ + yQ. 2 = 0 
 
 Eliminating P from equations (24) and (25) we have 
 
 (26) (Kp 2 + eP 2 (l-a~o-v) + Yp2(l- a -t- Y )2 = o 
 
 C2f, 
 
 at + 4 + v — H ( a +3 + y ) + v ( r. * 2 + y ) ' — >' y ( a + : + y ) j =0 
 P 2 ( a + 2 + y ) II- 2 y + v 2 — : ( 1 — y / — a ( 1— y ) + o: ,1 = C 
 P 2 ( a + 3 + y ) [ ( y-1 ) 2 + ( y— 1 5 ( a+ ? ) + a] =0 
 From which we see that the complete intersections consists of 
 
 70— circle 
 
 and 
 
 the 
 
 nodal c 
 
 ircle counted 
 
 t w 1 ' 
 
 and the inter 
 
 see 
 
 t i o n 
 
 of the 
 
 cubic with a 
 
 r uled 
 
 The existence 
 
 of 
 
 s e d 
 
 tics b a s 
 
 already been 
 
 seen. 
 
 Octic curves 
 
 can 
 
 b S 
 
 ob tained 
 
 by pass x n g a 
 
 cub ic 
 
 ■yclide 
 
 through a circle of the quintic and the straight line 
 
 A special class of nonics has been noted above. Others can 
 be obtained by passing a. cubic eyelids through a gsnsratic 
 
 circle. 
 
 Curves of order 10 can be obtained by passing cubic cyclid.es 
 through the straight line of the quintic. 
 
 Curves of order 11 occur as the general intersection of cubic 
 

 
 ■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1 '/ 
 
 c vc 1 ides with tbs quin tic eyelids. 
 
 It is apparent from the preceding that curves of inter sec t ioj 
 of surfaces with the quintic eyelids, of order lower than the 
 natural intersection, can be obtained by passing the intersecting 
 surface t fa r o ug the nodal circle, a gen e rating circle and a 
 straight line either singly or in combination. This reduces the 
 problem of finding a curve of any order on the surface to the 
 problem of expressing its order n in terms cf the modulus 5. 
 
 n = 5x - y 
 
 y may be made to take the values 1,2,3, 4. x is the order of 
 the required intersecting surface. It is e v idsn t t h e n t h at n 
 can always be expressed, in the manner required and hence that 
 a curve of any desired order can b-s found on the quintic . 
 
16 
 
 3= Minimal Lines on the O.uintic Gyclide . 
 
 Theorem 14. There are six pairs of minimal lines on the 
 quint ic eyelids . 
 
 ( 1 ) 
 
 ( 2 ) 
 
 f hen t h 
 
 (3) 
 
 » y* o 
 
 de r 
 
 he 
 
 condition 
 
 t hat 
 
 H ^ /-> 
 
 i i •_* 
 
 orrs 
 
 spending s 
 
 o he r 
 
 - A 
 
 ,o 
 
 n 
 
 1-A) -2 
 
 •!' a x 
 
 ■A 2 + 
 
 6 A 
 
 ►h rr <r\ v 
 
 i i ^ 
 
 + 9? 
 
 ■eq ui 
 
 red 
 
 condition 
 
 is 
 
 —A 
 
 r\ 
 
 ' ! • i 
 
 Y 2 *? I 
 
 7 3. 
 
 O 
 
 1-A 
 
 0 if. 
 
 Ud p 
 
 r\ 
 
 0 
 
 1— A Wr ( 
 
 \p 3 
 
 T i 
 
 T ? 
 
 \lf p «; 4 
 
 14 
 
 T i 
 
 (P p 
 
 CP 3 Cp 4 
 
 n 
 
 v 
 
 •andeni 
 
 ne q s are oi degree 2 in A and. the v; s of degree 
 
 7 in A. If th 
 
 O Oi \.r 
 
 x d a r.sio n 
 
 1 in A this equation is of deg res 
 carried out it is found that '■ -A : s a factor. He nee A=1 is 
 one root of the equation. 3ut the sphere corresponding to A=1 
 is the degenerate sphere A-8=Q and hence the pair of lines in 
 
 this particular case consists of the ling 
 A— 3 = 0 
 
 0! + f. ~ Y = O' 
 
 (4) 
 
 and the line at infinity in the clans 
 o' + ? + y — C 
 
 The other lines of the curve of contact are ‘the six pairs of 
 minimal lines. 
 
 The existence of the six pairs of minimal lines may also 
 be established in another manner . 
 
 Find the coordinates of the center of the sphere and set 
 uo the condition that the radius shall be equal to the distance 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4. Circles on the Guintic. 
 
 Pass a sphere through any generating circle of the 
 
 quin tic eye lids, e .g . through the circle 
 P - A* = C 
 a A 2 + 8 A + y = 0 
 
 ( 1 ) 
 
 where A is considered as fixed. Such a sphere is 
 
 (2) P - AG - u ( a A 2 + 8 A + y) = Q 
 
 This sphere cuts the quint ic cyciide in the sphero-circle and 
 a rest curve of orisr four which can be obtained as the inter- 
 section of the s p h ere w i t h ?. c u a. d r i c , 
 
 Trite the cyclide in the fern 
 
 (3) a (R 2 -A ) 2 + 8(R 2 ~A)(3 2 ~3) + y(3 2 -3) ? - C 
 and the so he re in t he f c r m 
 
 (4) S - R 2 ( I— A ) - - * Ac - u ( a A 2 + 8 A + y ) - C 
 Eliminating 3 2 from eq nations ( : } and (•:') we have 
 
 (5) 
 
 a f -4 - AS* u(a\ z -r A * y ) _j-| 2 
 
 + a r A — A 3* u( o A 2 e 8 A + v ) i r A~AS t u j q ; A 8 hz y l 
 
 ~1-A 1-A 
 
 v r A— AB+ji ( cAf_“3Ai: v )_q i 2 - ^ 
 
 1-A ~ J 
 
 B] 
 
 or 
 
 ( 6 ) C£ [ A ( 5-3 ) + u ( 0 : A 2 + - A + v ) ] 2 + 
 
 + 8 [ A ( A —8 ) + u ( 0 : A 2 + 8 A + y ) j | A-B + u ( $ A 2 •* 3 A + y ) J 
 + y [ A - 8+y ( a A 2 + 8 A + y ) ] 2 = 0 
 
 E x p a n d i n g a n d g r o u p i h g ' t e r is s w e 1 h a v e 
 
 (7) u 2 ( ot A 2 + 8 A -I- y ) 2 ( a + 8 + y ) + u ( a A 2 + -8 A + y ) [ 2 a A 1 2 -(• A f 1 ) * 2 y J ( A 
 
 + ( A — B ) 2 ( a A 2 + 8 A + y ) = 0 
 
21 
 
 This reduces to 
 
 ( 8 ) ( a A 2 + ? X+ y ) [ u 2 ( <xX 2 + ? X+ v ) ( a + 3 + y ) + u ( A - 8 ) ( 2 k X + ? + ? X 
 
 -t-fl-B) 2 ] = 0 
 
 (8) ( o: X 2 + 8 X + y ) K ' = 0 
 
 That is, the intersection of the sphere with the ouintic cyclide 
 consists of its intersection with the plane 
 <xX 2 + 3X + y = C 
 
 and with the quadric K r . 
 
 a n c 1 -c* i 
 
 J fi i- ^ -L ^ O : 
 
 any quadric through thi 
 
 auartic curve or. intersection 
 
 (10) K 
 
 v n. = 
 
 or e xp 1 ie i t ly 
 
 (11) (.4-5) 2 + u( A -8) (2a X + P+8A+2v)+ u 2 ( c: + 3 +y)( a X 2 + - A + y ) 
 
 — v l y 2 ( 1 — X ) — A + X b — ,u ( a X 2 + v X + y ) ] = 0 
 ;oef f icisnts of the various terms in the ouadrie (11) 
 
 7'h 
 
 contain X, u, v in different powers 
 
 table. 
 
 ndicated in the following 
 
 ;r ffl 
 
 X 
 
 ( 12 ) 
 
 y 2 
 z 2 
 xy 
 xz 
 y z- 
 x 
 V 
 z 
 
 const a n t 
 
 A > 
 
 ■2 1 
 
 up 
 
 A necessary and sufficient condition that a quadric break 
 into two planes is that every section z=c shall be a 
 
degenerate 
 
 That is, if the 
 
 22 
 
 conic. That is, if the equation be arranged in terms 
 of x and y the discriminant, which contains z, shall vanish 
 identically. 
 
 Let the equation of the quadric be, for example 
 (13) a 31 x 2 + 2 a 3 P x v + a. , , y 2 + 2 a 1s x z t2a 1A x + 2 a ? , y z + 2 a 9 A y 
 
 +a 3S z 2 +2a 34 z+a 44 = 0 
 
 This can be arranged in the form 
 
 (14) 
 
 a . i x 2 J -- a 1? x y 
 
 
 ( a , o.z 
 
 "i* 
 
 a , 4 ) x 
 
 
 + a 3 ? x y + a ? ? y 2 
 
 ~r 
 
 ( a p 2 z 
 
 -r 
 
 a 2 4 ) v 
 
 
 + (a,, 2 +a, Jx + 
 
 (a 
 
 p p z + a 
 
 P 4 
 
 ) y + ( a p ^ z 
 
 and the 
 
 d i s c r i m i n a n t i 
 
 
 
 
 
 (15) 
 
 
 3. 2 
 
 
 3. 
 
 ^ p Z 1 fi ^ 4 
 
 
 a i? 
 
 2 2 
 
 
 3. 
 
 2 3 z + a p 4 
 
 
 3.ijZ + S ^ 4 3 j ? Z 
 
 + a 2 
 
 ^ * s 
 
 s Z 
 
 2 +a~„z+a„ • 
 
 ^3 4 ^'4 4 
 
 Setting this 13 ' i ual to ’ and : i ' th 
 
 resulting equation in* power of z 
 
 g J \ Si 3 3. p O o ” ■/ 1 c 3 o •; ‘ O ^ n p ^ r; Ci o c "" - • - -j *5. p o ‘ Cl i «j v. n o c; ■“ d. a ^ c3. ^ rr / Z 
 
 x • _ - 
 
 c *3 C - - p - p -j- C, O O — ' ;1- P P p — . P O 
 
 r O i p <3. p p 44 w 1?44 - 1 1 ? ^ 1 4 ? 4 •'!!'•? 4 v 1 9 • 9 4 v 14 ^ 
 
 Then each poyncnaial coefficient must vanish identically. This 
 gives rise to three equations in A , q, v whose degrees, found 
 by making use of the table (12), are expressed symbolically by 
 [ A 6 q 6 v 3 J = C 
 (17) [ A 6 {i 5 v 3 } = 0 
 
 [ A 6 q 4 v 3 ] = C 
 
 three equations of degree 15,44,13 
 
 These 
 
 respectively, b are e 
 
 MM 
 
24 
 
 5. Special Quint ic OycIicI.es „ 
 
 Theorem 16. The quint ic cycoide is not, in General, 
 anal la£ma tic. 
 
 irs sfc ' to the Special Pentaspher ical Coordina 
 by means of the transformation^ 1 ' 
 
 2 A x = 2«. x t A ( x 2 + y 2 + z 2 -S 2 ) = RT6, x, 
 
 Ic if V ]r 
 
 A (x 
 
 + y « + 7, •- + 
 
 A 2 ) = -i.R2si,x 
 
 (1) 2 Ay = SP k x k 
 2Az = Oy k x k 
 
 the equation of the q air. tie eyelids becomes of the form 
 
 (2) 2u k x3 + 2u k4 xfx.> : .u 1 . )U x : ,X i1 x i = C i , 4 , k=l , 2, 3, 4, S . if/jfk 
 
 I n o r d s r 
 
 f o r 
 
 the s u r f a c s 
 
 anallagmat ic 
 
 it must! be 
 
 p o s s i b 1 e t o f i 
 
 7} j ? 
 
 n o r t h o gone 
 
 1 transformation of 
 
 such a nature 
 
 that there is 
 
 d o 1 
 
 east one of 
 
 t he f i v e v ar i ab 1 e s 
 
 Tf H n r» p» 
 
 /V I: 1 'w* 1J P vt- III b 
 
 only in even powers, since an inversion in pentaspher ical 
 coordinates with respect to a base sphere amounts only to a 
 change in sign of the one variable corresponding to that sphere. 
 That means that for one variable, say x^the third and first 
 powers must vanish. But this involves eleven conditions. In the 
 linear transformation on five variables there are twentyfour 
 essential conditions at our disposal. Cf these fifteen are 
 required to make the transformation orthogonal . ( z ^ Bence there 
 are only nine conditions open for choice and there are eleven 
 conditions to be fulfilled. Therefore a transformation of this 
 kind is not, in general, p os sib le . 
 
 (l) D a r b o u x : Geometric Analytioue p 385 
 
 (g) Boeder Introduction to K i g h e r Algebra p 154 note. 
 
Bb 
 
 There are, however, a few special quintic cyelides which 
 ire anallagmatic. 
 
 1. Consider the quintic eyelids generated b.y 
 x a 2 + y a + z = 0 
 
 ^ n 
 ■>' - 0 
 
 ( 1 ) 
 
 where 
 
 ( 2 ) r = x 2 + y 2 + z 2 — r 2 
 
 Q. = x 2 + y 2 + z 2 + r 2 = 0 
 
 so that the pencil of spheres is a concentric pencil with the 
 center the vertex of the double tangent cons. It is at once 
 apparent that the eyelids is symmetric with respect to the 
 
 origin. 
 
 Now 
 
 i f W 3 
 
 i n t 
 
 (3) 
 
 t_ 
 
 y 1 
 
 
 Q - 
 
 y ? 
 
 
 X = 
 
 y s 
 
 
 y = 
 
 y. 
 
 
 2 = 
 
 V 5 
 
 the equation of this quintic eyelids takes the form 
 
 (4) y a y I + ytJiSa + y s y§ = o 
 
 An inversion with respect to one of the base spheres amounts 
 to a change in sign of the corresponding variable. Hence the 
 c.yclide is unchanged by a. double inversion on the base spheres 
 y , y P . But this is merely the condition for symmetry with 
 respect to tbs origin. 
 
 As we have seen, it is, in general, impossible to find an 
 orthogonal substitution which will leave the equation of the 
 quintic eyelids with one variable occurring only in even 
 powers. In the special case which we are considering, a special 
 
transformation can be found 
 
 It will be noted that the equation is quadratic in 
 
 Consider 
 
 then. 
 
 a transf or mat ion 
 
 
 
 
 
 (5) 
 
 y% = 
 
 ay{ + by 4 
 
 
 
 
 
 
 .y? = 
 
 cyi + dyJ, 
 
 
 
 
 
 
 y a = 
 
 ey4 + 
 
 f .V 4 
 
 + §y.4 
 
 
 
 
 .¥4 = 
 
 by 4 + 
 
 . j y I 
 
 
 
 
 
 
 lyi + 
 
 m y ,! 
 
 + ny 4 
 
 
 
 A p p 1 y i n g 
 
 this transf or mat io 
 
 n to 
 
 e q u a t i 0 n ( 4 
 
 N 
 
 / 
 
 w e h a v e 
 
 ( 6 ) 
 
 (s.y4 
 
 + f y \ + g y I, ) ( a 2 y 4 
 
 2 + 
 
 9 Q W T7 f 77 I -L K 2 
 
 Ci wb ,y 1 .y p • o 
 
 vl 
 •' 2 
 
 2 ) 
 
 + 
 
 (by. 4 
 
 + jyj. + ky 4 ) (acy \ 
 
 2 
 
 (ad + b c ) y i y 
 
 7 
 
 9 
 
 + b d y 4 2 ) 
 
 + 
 
 (i.y4 
 
 + my J. + ny 4 ) (c 2 y 4 
 
 2 + 
 
 2cdy 4.y 4 + d s 
 
 y.4 
 
 2 ) - r, 
 
 J 
 
 A r r 2 . n g i n g 
 
 t his 
 
 -5 y-» y\ /p >7 O V» -r* /“V +* T 7 
 
 . 1 . LJ L_, v_/ .J w . ; *) 7 
 
 . > 'll 
 
 nd suppress! 
 
 ng 
 
 primes 
 
 facility in w r i t in'-, ; s h 3 v g 
 
 ( 7 ) v 2 i ( a 2 s + a c n + c 2 i ! y ,, +• f. n ? ? » /: -*-£ •- x ) y . + ( a 2 g + ack+c 2 n ) y r j 
 
 + y x y 2 [ (2abe+ (ad+bc ) h+ 5c dl ).y R + (2abf + (ad+bc )„j + 2cdm )y„. 
 
 + (2abg+ (ad + oc )k+2cdn )y 5 ] 
 
 +y | [ (b 2 e+bdh+d 2 l ) y r + (b 2 f +bd j +d 2 m )y 4 + (b 2 g+bdk+d 2 n )y = ] = C . 
 Introducing the condition that the y x y P _ term shall vanish gives 
 
 (8) 2a be + (ad + bc)h + 2cdl = 0 
 
 2abf + (ad + bc)j + 2cdm = 0 
 
 2abg + (ad + bc)k + 2cdn = 0 
 
 But from the conditions for orthogonality cd=-ab and (8) reduces 
 
 to 
 
 ( 9 ) 2ab(s-l) + (ad+bc )h = 0 
 2ab(f— m) + (ad + bc ).j - 0 
 2ab(g-n) + (ad+bc )k = 0 
 
 From ( 9 ) follows immediately 
 
 (10) _2ab_ _ _h_ = _i_ - _k_ 
 
 1 — e 
 
 ad+bc 1-e m— f n-g 
 
 Squaring and applying a theorem of proportion we have 
 

 
 
 
 
 
 
 
 
 
 
 . 
 
 
 
 
 
 
 
 
 . 
 
 
28 
 
 b. inverse of Ruled Cubic. 
 
 ( 1 ) 
 
 msider a ruled cubic generated by 
 «X 2 +pA+y=0 
 a - A b = 0 
 
 where a,?,v,a,b are linear in x,y,z. The equation of the cubic 
 
 i s 
 
 ( 2 ) 
 
 2 -i. O 
 
 cao + vo 
 
 It involves no loss of generality to assume a , ?, v homogeneous 
 
 i n x , y , 2 . 
 
 Now apply an inversion 
 
 (3 ) x : x 1 = y : y 1 = 
 
 2 • v 1 — jj» • y ! 2 y ! 2 7; ! 2 — 2 .f. y 2 -f. g 2 * r* 
 
 with respect to a sphere of rsdius r and center at the origin. 
 Such an inversion transforms ~~y plane into a so here passing 
 through the center of i nversi c r and in oar ici lar , if the plane 
 passes through the center of inversion it is transformed into 
 
 itself » 
 
 Since «, 8, y are homogeneous in x,y,z they are planes 
 
 passing through the center of inversion and are therefore 
 
 unchanged. The two planes a,b are, however , transformed into 
 
 spheres P and Q, Then the surface into which the cubic is 
 
 transformed is generated by 
 
 7 . a 2 + 8 a y 0 
 
 D __ } — O 
 
 where P and 2 are spheres cassing through the origin. 
 
 But this ia a quin tic eyelids with the center of the 
 double tangent plans on the nodal circle. 
 
 The doable line of the ruled cubic which is 
 (5) a = o = 0 
 

?Q 
 
 -O t/' 
 
 is transformed into the intersection of P and Q and hence is 
 the nodal circle of the quintic. But since both spheres oass 
 through the center of inversion the nodal circle passes through 
 that point. Since through the original center of inversion on 
 the cubic there passes a generator of the cubic, then there 
 exists a straight line on the quintic. 
 
 Theorem IS. - - . enei 2 tin circle t is s Dedal quintic 
 eyelids passes through the center of inversion and one other 
 point on the nodal circle. 
 
 As a further particularization the center of inversion 
 might be assumed as a poi . the nodal line of the ruled 
 cubic. In that case t n s n o i 3 . 1 line as a ? a i r a s t r s i g h t 1 i n e 
 after the transformation 7 hi is the planes, of the cone are 
 transformed into spheres and the new surface has an equation 
 
 (6) Pa 2 + Qab + Mb 2 - 0 
 and is generated by 
 
 (7) 
 
 PA 2 + QA + H = 0 
 
 a - Ab = 0 
 
 But this is a quartic cyclide with a nodal line. 
 
 Inasmuch as the auartic cyciidss have been rather fully 
 treated in the literature of the theory of surfaces a further 
 discussion of this particular surface qculd 0 e out of place 
 
 here , 
 
 Again, if the center of inversion be assumed as any point 
 in space, the cubic will be transformed into a sextic cyclide 
 
 ( Q 'S p p 2 j. P P P + p p 2 = 0 
 
 V / *- 1 - p ' - 4 ^ ^ 
 
31 
 
 II. General Cyclide. 
 
 1. Projective Descriotion. 
 
 Proceeding' from the method of defining the auintic 
 cyclide we shall define. a general cyclide as the surface generat- 
 ed by the pencil of spheres 
 ( 1 ’i p _ x o = r 
 
 and the protectively related planes of the developable 
 (3) cc 0 A ?1 + o: 1 A n ' _1 + a ? X n ~~ 2 + ... J >- o: n _ t A + a ?7 = C 
 
 and the equation of the general cyclide is tree 
 
 (3) ff. 0 P n + c/ 5 P n - 1 C - + ... + <X„ — aPQ*- 1 + = 0 
 
 This surface has the sphere-circle at Infinity and the 
 radical circle of t h e p e n c i i. c f s p h ares as a r. u 1 tide c u r v s 
 of order e .a d t h ere f c c e c e fc c - s n e ; , t cl a ss c f s u r f aces 
 
 si oraer t n + .; r.a v i r ; - ; r 
 
 y ■ c J 
 
 ' ' - 
 
 Theorem. The developable surface (2) is doubly tangent to 
 the genera l cyclide (3). 
 
 Any two planes of the developable surface meet in a line 
 which intersects the surface in 3n+l points. Each plane cuts 
 the cyclide in a circular generator. Put as the planes approach 
 coincidence the two circles approach coincidence and the line 
 approaches a generator of the developable. Then the four points 
 in which the line cuts the two circles come together in pairs 
 so that the generator of the developable surface is tangent 
 to the cyclide in two points. 
 
 Again, a generator of the developable is given by the 
 equations 
 (4) 
 
 a o A ” + a a A ” ~ 1 + v P X n ~ 2 + ... 
 na 0 A n_1 + (n-l)a 1 A n ~ 2 + ( n-2 ) a ? A n ~ : 
 
 ' K n — a o 
 
, 32 
 
 Eliminating a n from the first of these equations and the equation 
 of the eyelids (3) we find 
 
 (5) (P-AQ) [oi 0 F rl ~ 1 +(a 0 A + o: 1 )P”~ 2 9+ . . . + (a 0 A n “ 1 + . . , + a n _ 5 )Q n ~ 1 ] =0 
 Then eliminating a n _ 4 from the second equation of (4) and (5) 
 we have f i n a 1 1 y 
 
 ) = 
 
 where i (a 0 , a a , . . . , h, n , A ) is a function, homogeneous of degree 
 n-2 in P,Q and linear in a 0 , a 1; . , . . This equation states that 
 the generator of the developable surface cuts the 
 two double points which lie on the sphere ?-A0=0 and in 2n— 4 
 points which lie on a eyelids of order in— , for the second 
 factor of (Sf is of the same form as (3) but of degree 4 less. 
 
 (6) ( P-.X9 ) 2 f ( a o , a 3. , . . . , P, Q , A ; = 
 
 s t a t e s 
 
 t h a t 
 
 c y c 1 i c 
 
 is in 
 
 a n d i n 
 
 2 n— 4 
 

2. Systems cf Curves on the General Cyclide. 
 
 I ! he ores 20. guery generatin'-* plane of the tangent develop 
 able outs the general cyclide in a circle and a rest curve 
 of order 2n-l which lies on a cyclide of order 2n-l . 
 
 The proof of this theorem is at once evident from the 
 
 * 
 
 examination of equation (-5) of the preceding section, which 
 can be written in the form 
 
 Cl) CP—? o ) f R C8 — 1 - 3 pn — S'-. + + a fan— i,— r - 
 
 the second factor of which is of the same form as i he equation 
 of the general cyclide (3) of the last section but lower in 
 degree b y 2 . 
 
 Theorem 21. Any so here cits the ■■■■ enerol eye line in o curve 
 o f or der 2n + 2 -o n i c h c a n b e p r o :} e c l - o. i ;■ .• i o ■ • p i a n e c ,• • v e o f 
 o r a e r 2 n + 2 a n d ? e n u s n . 
 
 The equation o f the general cyclide may be written in the 
 
 f o r m 
 
 (2) a o (^ 2 -A) ?I + 0! 1 (R 2 -:5)»- i (a 2 -3) + . . . + a n ( 3 2 -3 )* = 0 
 
 ana any sphere 
 
 (3) R 2 - 2 = : 
 
 T hen eliminati n g 1 2 o e t w sen ( 2 } a n d ( 3 ) . . e b a v s 
 
 t 0 (C-A )»+a 1 (C-A)«” 1 ( ;-F )+. . .+a n _ a ( - ( 0-3 )»“i+a n ( C~B) n 
 
 ~ \J 
 
 But thisiis the equation of a ruled (n+1 )-ic, generated oy the 
 planes of c he levs iopab le 
 
 (5) <x 0 A ?I + a- L X n ~ 1 + ... + a n _ 1 A + a. n = 0 
 and the pencil of o lanes 
 
 (6) (G-A) - A (0-6 ) = 0 
 

The rulsd surface has an n-f old 11ns 
 (7) C-A - 0-3 - 0 
 
 34 
 
 The sobers cuts the general eyelids in the curve in which it 
 cuts the ruled (n+1)— ic. 'hen the curve of intersection has 
 2 n-f old points. If this curve is projected on a plane from 
 any point in the n-f old line of the ruled surface, the result 
 is a curve of order 2n+? with a 2n-fold point and n ordinary 
 doubly points . (corresponding to the n generators of the ruled 
 surface passing through the point of the n-f old line used as 
 a center of projection). 
 
 The 2 n-f old point counts for 
 in(|n-l) = 2p2 _ n 
 
 ordinary double points. Then this with the n double points 
 makes a count of 3 n 2 double points . but the maximum number of 
 possible double points is 
 
 |2ni5-l2iin + 2-2) = ? n 2 n 
 
 Then the deficieatcy is 
 
 2n 2 + n — 2n 2 — n , 
 
 a result which we have already seen in the ouin tic. eyelids 
 
 where n=2. 
 
 By a treatment exactly analagous to that of I, 2 , a po 1C— 1 h 
 there is developed the 
 
 Theorem 22. The general eye tide can be Generated in a trivia 
 infinite number of ways by a recoil of (2n~l)-ics and a pencil 
 of concentric spheres. 
 
J5 
 
 3. Locus of 
 
 the Centers 
 
 of the 
 
 Generating 
 
 ^ "I V»|T» 
 
 les . 
 
 Theorem 2 
 
 . The locus 
 
 of the 
 
 centers of 
 
 the 
 
 general in g 
 
 circles of t h e 
 
 general eye 
 
 lide of 
 
 order 2n + l 
 
 is a 
 
 space curve 
 
 of order 2n+l. 
 
 Consider the generating plane 
 
 ( a ) « 0 a 71 + v .^ 71 - 1 + = o 
 
 or 
 
 (2) xcp-t + + zo, + cp 4 = 0 
 
 where the <p 1 s have the same s'* gri^'ieanee as in the coin tic 
 eyelids except that they are here of degree n in A. The corre- 
 sponding sphere of the pencil is 
 
 (3) ( 1-A ) R 2 - ?'!f t x - ?p 5 y - - 2»u = 0 
 
 where the 1 s have precisely the T.eanin- 
 
 O U 1 n t, 1 
 
 The line from t hs 
 
 inter of t he sobers perns ndicular to the 
 
 corresponding ilane is 
 
 ( ■ ) Alz^izzTi = ilz^llzl a - izz^lizi* 
 
 ?i T ? <P 3 
 
 The center of the corresponding circular generator is the 
 intersection of the line (45 with the plane (2). Solving for 
 the coordinates of the point of intersection in terms of A -we 
 
 have 
 
 ( 5 ) c x 
 
 py 
 
 pz 
 
 P t 
 
 ( 1-A ) 33 1 cp 4 + © . , ( © 5 ;y ? + cp n )-cp , ( cp f + © | ) 
 
 (1-A )co ? © 4 + y ? + )“®j» (Ti + *I ) 
 
 (1-A ) <P r, © 4 + <p o (© i w i + ® 2 u/ ? )— cp, (of + cp| ) 
 (1-A ) (cpf + cpi + cpf ) 
 
 But these equations are of degree 2n+l in A and hence the curve 
 is a space curve of order 2n+l. 
 
 The locus of the lines perpendicular to the generating 
 planes from the centers of the corresponding spheres is a ruled 
 

 r» o 
 
 . V w 
 
 of 
 
 crdei 
 
 ? R + 
 
 1. 
 
 
 
 
 
 
 
 Co 
 
 nsi 
 
 d e r any 
 
 plan 
 
 a. 
 
 
 
 
 
 
 (6) 
 
 
 ax + 
 
 by 
 
 + c z 
 
 a. 
 
 d 
 
 — o 
 
 — ^ 
 
 
 
 
 is 
 
 no 
 
 loss 
 
 5 Of 
 
 c? 0 n 
 
 erali 
 
 t y in 
 
 choos 
 
 ing 
 
 c = 1 , T 
 
 wee 
 
 n e 
 
 q uat ion 
 
 (e) 
 
 and e 
 
 q uat i 
 
 on (4) 
 
 we 
 
 have 
 
 (?) 
 
 
 (1— A 
 
 ) a cp P 
 
 x + (1 
 
 -a; 
 
 )(© 
 
 p. + b cp p 
 
 )y+dcp ? 
 
 + CO p 1 
 
 „ — o: „ v; - 
 
 
 
 c i — a ; 
 
 ) (a© 
 
 l+?3 
 
 )XH 
 
 1 
 
 — A ) b cr. 
 
 
 + a j 
 
 A' p - Cp p © 3 
 
 (7) 
 
 t h 
 
 ere r s s u 
 
 It'S 
 
 
 
 
 
 
 
 (S) 
 
 
 ox = 
 
 d© a 
 
 + cp J \<t 
 
 !>■ - 
 
 P ?. ? 
 
 i + b ( © 
 
 t ©p-© ? 
 
 
 
 
 
 py = 
 
 d cp P 
 
 + © P \n 
 
 , r 
 
 o r w 
 
 p + a ( cc 
 
 o VJ x — <p 5 
 
 \|/ 2 ) 
 
 
 
 
 ot = 
 
 -(1 
 
 — A ) ( 
 
 cp 3 - 
 
 r 0 0 
 
 1 + b cp ? 
 
 ) 
 
 
 
 ea 
 
 uat 
 
 ions 
 
 are 
 
 of 
 
 dec 
 
 Ire 
 
 e n + 1 
 
 i n A 
 
 a nd 
 
 hence 
 
 fhess equations are or degree n + j in a ana nence t ns surtace 
 is of order n+1. The sane plane (6) intersects the oianes of 
 the developable surface in a class curve of class n » fence 
 there is an (( n, n+1 ] —correspondence between the points of this 
 curve of order n+1 and the tangents of the class n-ic. Tut 
 in such a correspondence there are n+(n+l) coincidences and 
 hence the locus of the centers of the circular generators 
 .meets this olane in 2n + l ooints and is therefore of order 2n+l. 
 
4 . Min 
 
 Cyclide . 
 
 
 ions ider tfee condition that a sphere of the generating 
 
 pencil is tangent 
 
 to t 
 
 he corresponding 
 
 plane . 
 
 
 1-A 
 
 0 C 
 
 V?/ 1 
 
 
 
 
 o 
 
 1-A 0 
 
 1/? 
 
 
 
 
 (1) o 
 
 0 1-A 
 
 V.-9 
 
 
 
 
 VI/ 1 
 
 \’J p VJ r. 
 
 \JJ M 
 
 T 4 
 
 
 
 
 CD 2 Cp 3 
 
 
 
 
 
 This is 
 
 an eouati 
 
 on of 
 
 degree 2n + 3 in A 
 
 since the cp 
 
 1 s are of 
 
 degree n 
 
 and the 
 
 ■■■11 ' 3 0 
 
 f degree 1 . But w 
 
 hen the dete 
 
 r m i n a n t is 
 
 expanded 
 
 it turns 
 
 cut 
 
 that 1-A is a fac 
 
 tor sc that 
 
 A = j is 
 
 one root 
 
 of the e 
 
 0 U 3 t i 
 
 on , 
 
 
 
 Ther 
 
 e for e one 
 
 p a i r 
 
 of lines is r e a 1 
 
 , name L . the 
 
 line 
 
 (2) 
 
 (X o ~ C! j 
 A - B = 
 
 + a ? 
 
 Oi 
 
 w 
 
 + . . . + a n = C 
 
 
 
 and the 
 
 line at i n f i n i 
 
 tv in the plane 
 
 
 
 (3) 
 
 a o + a a 
 
 + a: p 
 
 + . . . + a n = 0 
 
 
 
 Corresponding to the 2n + 2 other solutions of equation (1) 
 there are 2n+2 pairs of minimal lines on the general cyclide. 
 
 If we set ud the condition for tangency as in the case of 
 the auintic cyclide we reach the condition 
 
 (4) [® i¥ a +<p ? ¥ ? + ¥nW<,+ ( 1— A ) Cp 4 3 2 = [<pf + tp| + cp|3 [v/f + u;§ + u;f + j 
 
 and this, we see at once, is of degree 2n+2 in A. 
 
 I‘he result is in agreement with the number of minimal lines 
 found on the quintic cyclide in which n=2. 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
3S 7 
 
 . Further Generalized Cyclides. 
 
 Consider the surface generated b .y 
 
 (1) a 0 (A L ,M,) 7I + a 1 (A,.u) n-1 -i-...c( n = 0 
 
 where A and u are connected by a rational equation 
 
 (2) R n (A,u) = 0 
 and 
 
 (3) F - AO. = 0 
 
 Equation (l) represents a general alg'eoraic developable surface. 
 Each plane corresponding to a value of A cuts the corresponding 
 sphere of the pencil (3) in a circle and the totality of such 
 circles forms the system of circular generators of the eyelids. 
 
 If jj — 0 we have the particular case which has been discussed in 
 the preceding pages. 
 
 The general case defined by equations (1),(2) and (3) 
 has .yet to be considered in detail. 
 
 Again, consider the surface generated by 
 
 (4) P 0 A n + p,A r ‘ -1 + PpA*- 2 + ... + F n = 0 
 
 where 
 
 ( 5 ) P . = R 2 - A . 3 2 = x p + y 2 + z 2 ' . linear in x , y , z . 
 
 and the pencil of spheres 
 (e) 0, - AQ ? = c- 
 
 This surface is a general eyelids of order 2n+2 with the equation 
 
 (7) P 0 Q« + 
 
 -i- P o 0 n~ i + p nn = 
 
 ~ Yl i - i p - ? 
 
 and should afford a good many interesting properties , and would 
 be worth invest i gat i n g . 
 
 A still further generalization would be to let the o: 1 s of 
 equation (1) represent spheres rather than planes. 
 

 
 
 
 - 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
VITA 
 
 Harvey Pierson Pettit, the author of this thesis, was born 
 April 25 1883 at Cbadrcn, Nebraska . His elementary education 
 was received in the public schools of Michigan. He was 
 graduated from Jackson High School, Jackson, Michigan in 1810, 
 and from Kalamazoo College in 1814 with the degree of A, 6, 
 During the four years, 1814-1818 he was teaching mathematics 
 in the high school at Holland Michigan. He was Fellow in Math- 
 ematics in the University of Kentucky 1816-18 receiving the 
 degree of M.A. in 1818 from the University of Kentucky. Ke 
 
 was an assistant and graduate student in Mathematics at the 
 University of Illinois 1818-1822. He is a member of the 
 Mathematical Association of America, Phi Seta Kappa, and 
 
 Xi 
 
 u a t< * 
 
 t r : S ODDOF 
 
 ^ y 
 
 i n g 
 
 h a n K s 
 
 tO t hi 
 
 members cr l r: ? 
 
 :• 3 C : as worked, and ir. 
 
 particular to Prof. Arnold Emch, under whose direction this 
 thesis has been prepared, for his kindly counsel and inspiration.