698 
Dll. J. CASEY ON CYCLIDES AND SPHEEO-QUABTICS. 
the four common generating spheres be a, (3, y, e), then the nodes of the four enveloping 
binodals are the four pairs of inverse points (a, /3, y), (a, y, &), (a, (3, l), ((3, y, h). 
299. If a sphero-quartic be common to three cy elides, each pair must have another 
sphero-quartic, and the spheres through these sphero-quartics are coaxal. 
300. W, W' are two cyclides having a common sphere of inversion U ; it is required 
to find the locus of the pole points of the generating spheres of W' with respect to W. 
Let W, W' be reduced to their canonical forms, 
W-^aa 2 + b(3 2 -f efi + do 1 , W'= a' a 2 + b'[3 2 + c'f + dV. 
Now let («', (3', y', ci') be a pair of inverse points with respect to U, then the polar sphere 
of (a!, j3', y', S') with respect to W is 
act! a + bj3'(3 -f - cy'y -j- fiS'S — 0 ; 
and the condition that this should be a generating sphere of W' is 
«V 2 , b q 3 12 c 2 y' 2 d q S' 2 A 
-r+-f+-r+T-=°: 
and omitting the accents, we have the locus required, 
(176) 
301. If we denote the cyclide (176) by "W"=0, we see that the equations of the focal 
quadrics of W, W', W", in tangential coordinates, are 
a\ 2 +bfi + cv 2 +df =0, 
a , k 2 +b'fi+c'v 2 +d , f= 0 , 
2 , 2 , c2 2 A . 
a' x +7 V - 0 ’ 
and the third is the reciprocal of the second with respect to the first. Hence we have 
the following theorem: — If W, W', W" be three cyclides having a common sphere of 
inversion U, and F, F', F" be the focal quadrics of W, W', W" corresponding to U, then 
if F" be the reciprocal of F' with respect to F, W" will be the reciprocal of W' with 
respect to W. 
302. Since the reciprocal of a circle in space with respect to a cyclide is another circle 
in space, hence, if a variable circle move along three circles, its reciprocal will move 
along three circles reciprocal to the former ; so that the reciprocal of a cyclide described 
by the motion of a variable circle is another cyclide described by the motion of another 
variable circle. This corresponds to the theorem that the reciprocal of a ruled surface 
is a ruled surface. 
303. If a pair of inverse points move along a fixed sphere , the locus of the pair of 
inverse points common to their polar spheres, with respect to three cyclides having a 
common sphere of inversion U, is a cyclide of the sixth degree, having U for a sphere of 
inversion. 
