DR. J. CASEY ON CYCLIDES AND SPHERO-QXTARTICS. 
637 
the focal sphero-quartics of such a cyclide must have a cusp , namely , the inverse of the 
point at infinity on the focal parabola. 
Conversely, if the sphere of inversion a of a cyclide be an osculating sphere of the focal 
quadric F, and if the whole system be inverted from the point of osculation, the sphere a 
will invert into a principal plane of the quadric into which the cyclide inverts , and the 
sphero-quartic in which a intersects F will invert into a parabola. 
116. Since tlie intersection of a focal qnaclric of a cyclide with the corresponding 
sphere of inversion gives a line of foci of the cyclide, then, if the cyclide be 
cia~ -\-bfi 1 cy 2 dh 2 , the focal quadric will be a/i -j- by/ -f- a / -}- df ; and if the sphere of 
inversion be given by the equation IP = a 2 +/3 2 -f y 2 + cfi, then the line of foci will be 
given as the intersector of the two surfaces in tetrahedral coordinates, 
h.fi. 
— L— — 0 
b + d ~ U ’ 
F ! +J ! +3 2 +w 2 =0, 
and therefore the quadric in tangential coordinates 
ax? Z//x- cv Q - dp _ 
a-\- b + k' c-\- k' d + k 
passes through the line of foci of aa 2 -\-bf 2 -\-cy' 1 -\-d}i l . Hence it follows that the cyclide 
i$i+m +&+£*= 0 ( 94 ) 
denotes in general a cyclide having one focal sphero-quartic in common with 
aa 2 -f- bf -f- cy- -f- dl 2 — 0 . 
117. In art. 33 we have seen that a cyclide given by the canonical form 
aa 2J rbf-\-cy 2 -\-d'b 2 -\ r ez 2 =i) may be written in five different forms; and by the last 
article we see that to this cyclide correspond five different systems of cyclides, each 
system having one sphero-quartic of foci common with it. These systems are given by 
the equations : 
(«-6)/3 2 , (a-c)y 2 , («-d)S 2 \a-ey 
a — b + k a-c-yk ' a — d-\-k ' a — e-yk ‘ •••••• [dOJ 
p-c)y 2 , (b-d)V f b- e y (b-ay _ 
b-c + k 1 ~i~b-d + k' ' b— e-yk! ^ (b-a)-yk' 
(96) 
'{c— d)h- , (c — e)P ( c—ay (c—b)(3- _ 
c—d+k " ' c-e-yk" ‘ c—a + k" ' c—b-y k" 
(97) 
{d-eY , {d-ay ( d—b )/ 3 2 {d-e)y~ 
d-e + k!" -rd~a + k'"^~d-b + kl"~ ] ~d-c + k"' ‘ ‘ W 
{e— a)ol {e—b)[ 3 2 (e— c)y 2 ? (e—d) S 2 
e — a + k"" 'c—b-y k"" ‘e-c-h k" n f e — d+ k ,fr ' ~ 
( 99 ) 
In these equations the Id s may have any value. 
4 s 2 
