DE. J. CASEY ON CYCLIDES AND SPI1EEO-QTJAETICS. 
625 
equation of the inverse spliero-quartic will be of the same form ; and in particular if the 
sphero-quartic be of the form aa 2 + ^/3 2 + ey 2 -|-f^"=0, the inverse spliero-quartic will be 
of the same form. 
82. We have seen that the general equation of a sphero-quartic can be written in the 
form 
a cc 2 -fi hf + cy 2 -f- dh 2 = 0, 
where a 2 -j- (3 2 J r y 1 -j- o 2 = 0 is an identical relation, and the circles are mutually orthogonal. 
This is also evident otherwise ; for the given form contains fifteen constants, namely, 
bilics if li 2 —a 2 — b 2 , kr—a 2 — c 2 ; two of the umbilics of contact are in the plane of (xz) and are real, and two in 
the plane ( xy ) and are imaginary. Now to find the equation of the cyclide that will have the ellipsoid and 
sphere for focal quadric and sphere of inversion, that is, for E and J ; the perpendicular let fall from the centre 
of J on any tangent plane to E is evidently equal to V a 2 cos 2 a. + b 2 cos 2 (3 + c 2 cos 2 y — S g ; but if 0 be the 
centre of J and T the foot of the perpendicular on the tangent plane, and if the points P, P' be taken so that 
OT 2 — TP 2 =OT 2 — TP ,2 = ^ , then P, P' are points on the cyclide ; and denoting OP by ^ and OT by p, we get 
2 P! = w +i, 
or. 
| V a 2 
cos 2 a, + b 2 cos 2 /3+c 2 cos 2 y — 
lik COS i 
b 2 C 2 
Hence, since cos a, cos /3, cos y are the direction cosines of OP or p, we get the equation of the cyclide 
. — „ , , 2hlcx b 2 c 2 
2 V ccx~ + h 2 y + cz — x- + y 2 + 2 " 4 — - — + 
or, denoting the second side of the equation by C, 4(a+ + hhf + e 2 z 2 ) = C 2 . 
Y. If in the equation 4(a+ + ¥y 2 -j- c 2 z 2 ) = C 2 , we put y=0, z=0, we get 
( 2 2 Tikx Fc 2 . 0 V , . 2hkx b 2 c 2 0 \ n 
f x 2 + + — + 2 ax ( x 2 + q — - ~2ax =0 ; 
\ a a 2 J\ a a 2 ) 
21 ilex , b 2 c 2 
4 hlc 
and the four values of x in this equation being denoted by x v x. 2 , x 3 , x 4 , we get x 1 -j-x 2 -j-x 3 +x 4 = — - 
U 
Hence it follows that the centre of the ellipsoid ^-+^-+^-=1 is the centre of the cyclide. The values of 
a- b“ c- 
x v x 2 , x 3 , x 4 are easily seen to be given by the equations 
_ , , , hlc . 
Ob 1 — CL -j- / 1 — iC — y 
x 0 = a-h + lc- 
, 7 . 7 . M . 
- Cl -|“ h tc y 
a 
a? 4 = —a — li — lc — 
lik 
a 
lik 
Hence if we had taken the centre of E for origin of coordinates in the equation of the cyclide, the four values 
of x would be given by the system of four equations, -ka + h + lc—Q. 
YI. The equation of the cyclide referred to the centre of E as origin is given by the following symmetrical 
equation : 
(+ + y 2 + z 2 ) 2 - 2x\a 2 + 7r + k 2 ) - 2 y\a 2 - ¥ + 1c 2 ) - 2z 2 (a 2 + Ir - lc 2 ) 
4- Sail kx + (« 4 +7i 4 + 7r 4 — 2 a 2 h 2 — 2h 2 k 2 — 2k 2 a 2 ) = 0 . 
The sections by the planes y—0, z= 0 are evidently the two pairs of circles 
(x -\-k)- + z 2 — (a-\-hy, (x — ky +z 2 — (a — If, 
(x + hf + y 2 — {a + If, (x - 7t) 2 + y 2 — {a — k ) 2 ; 
