612 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
where (12) stands for the invariant 
(1 — R) — \/ (1 — S')(l — S") of two conics. 
49. If we eliminate k between the equation 
S< — L+/r(S*— M), 
and the discriminant, 
k~ tan 2 g" + 2 k tan g' tan g" cos C -f- tan 3 g' = 0, 
we get 
(S* — L) ? tan 2 g"— 2(S“ — L)(S“ — M) tang>' tang>" cos C + (S*— M) tan 2 ^'=:0. . . (53) 
This is the equation of the limiting points of the two circles S J — L=0 and S 1 — M = 0; 
and they evidently correspond to the vertices or points of intersection of the two pairs of 
lines which can be drawn touching the conic S through the points of intersection of the 
conics S- — L and — M, with their common chord L— M. Compare art. 8, equations 
(11) and (12). 
Cor. The equations of the pair of points diametrically opposite is got by changing the 
signs of L and M in the circles S=' — L and S* — M. 
50. We may get the equations of thedimiting points otherwise. Thus, if cos a', cos (3', 
cos y ' ; cos a", cos [3", cos y" be the direction cosines of the planes L and M, then, when 
we write the equations of the small circles in the form S^— L = 0, S* — M = 0, we must have 
L = sec g' (x cos a! -j -y cos j3' + z cos y ’ ), 
M= sec g"(x cos cc" -\-y cos (3 " -\-z cos y"). 
Let, then, the circle S= — L-f-d^S 1 — M) = 0 be denoted by 
— secr(# cosX+y cos^ + z cos i>)=:0 ; 
and if this reduce to a point, we must have secr=l. 
Hence, comparing coefficients, we get 
(1 -}-/:) cos A = sec g' cos a' -\-k sec g" cos a", 
(1-f-d) cos (Jj= sec g' cos (3' -\-k sec g" cos (3", 
(1 -\-k) cos v — sec g 1 cos y' -\-k sec g" cos y" ; 
square and add, and we get, after a slight reduction, 
k 2 tan 2 g" -{-2k tan g' tan g" cos C+ tan 2 g' = 0, 
the same as before. 
51. We can now, from the results proved in ‘ Bicircular Quartics,’ write out at once 
corresponding ones for three small circles on the sphere. Thus, from the equations of 
the four conics J, J', J", J'" orthogonal to three given conics, S — L 2 , S — M 2 , S— N 2 =0, 
we can write out the equations of the circles cutting three circles orthogonally. Thus if 
the circles be S^— L, S* — M, S* — N, their spherical radii g', g", g" 1 ; direction angles of 
the planes L, M, N he a', /3', y' ; a", (3", y" ; a'", [3"', y'", the orthogonal circle J is the 
