DK. J. CASEY ON CYCLIDES AND SPHEEO-QTIAETICS. 
701 
whose centre moves along a great circle cuts a given circle J orthogonally, it will pass 
through two fixed points ; these fixed points are the limiting points of J and the great 
circle ; or if we denote, as Dr. Salmon does, the equation of a great circle by an equation 
of the first degree in x y z , say L=0, and the circle J by the equation S"— M=0, then 
the two limiting points will be given as those for which the discriminant of (S J — M-f-A'L) 
vanishes. These points will be the pole points of the circle Aja-f yo/3 + hy— 0 with respect 
to the quartic (a, b, c,f, g, lija, 0 , y) 2 =0. 
Cor. 1. The great circle. (181) is the polar of the centre of C 1 =‘X l a J cg^(^ J rv l y with 
respect to the sphero-conic whose tangential equation is 
Ax 2 + Bjmt -j- Cr T 2 H Xg> -f - 2 F (mv 4- 2 G vX = 0 , 
where, as usual, A = bc—f ~, 'B = ca—g 2 , &c. 
Cor. 2. If two circles be such that one of them, A, passes through the pole points of B, 
then, conversely, B passes through the pole points of A. 
310. If (A, 0', y') be the cyclic coordinates of a pair of inverse points, that is, the pair 
of points given by the system of circles 
a , fc 
7 > 
a', 0 
and a", 0 ", y" the cyclic coordinates of another pair of points, then la! -f ma", / 0 '-fw 20 ", 
ly'-Amy" will be the coordinates of a pair of points coney clic with them; and if these 
satisfy the equation of the sphero-quartic, which we may denote by Q, we shall have 
Z 2 Q , + 2 / r }nP+m 2 Q" = 0 (182) 
Now, if P = (J the circle through («', 0', y'), (a", 0", y") meets the quartic in two pairs of 
points which are harmonic conjugates with respect to the two pairs (a', 0 ', y'), (a", 0 ", y") ; 
but P is 
( a '^+^V ,+y V) Q,, ‘ 
Hence the equation of the polar circle of the points (a!, 0 ', y') is 
(4+' 3 'Jw'4)Q=° (182 a) 
Cor. 1. From this article we have evidently the following theorem : — If through a pair 
of inverse points a', 0', y we describe any circle Z cutting the polar circle of (a!, 0', y) in 
a pair of inverse points ( a 0", y"), Z will cut the quartic in two other pairs of points, 
such that the four segments made on Z by (a', 0', y'), (a", 0", y n ) and by the quartic are 
harmonic. 
Cor. 2. If (a r , 0', y) be a pair of points, the generating circle which touches Q at 
(«', 0 ', y) is 
( a '^+^ + V^)^=° (183) 
Cor. 3. If through a pair of inverse points we describe two generating circles of the 
5 c 2 
