DR. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
Gil 
their points of intersection. Forming the discriminant, we get 
(1 - S")£ 2 + 2(1 - R)/r + (1 - S') = 0, (48) 
where S', S" denote the results of substituting the coefficients of a 1 , y , z from the planes 
L and M in the point sphere S (.r ! -l-y 2 -|-z 2 =0), and 11 the result of substituting the 
coefficients from one of these planes in the equation of the other. Hence if q" be the 
spherical radii of the circle S* — L and S^— M, we have 
1 — S' = — tan 2 q', 
1 — S"= — tan 2 q", 
1 — R = — tan q' tan q" cos C, 
where C is the angle of intersection of the circles. Hence the quadratic (48) becomes 
tan 2 q" # 2 +2(tan q' tan q" cos C)#-j- tan 2 q'=0, (49) 
and the discriminant is 
tan 2 q tan 2 q" sin 2 C ; (50) 
and this is what corresponds, in the geometry of two small circles on the sphere, to the 
invariant of two conics, 
(1 — S')(l — S") — (1— R) 2 . 
See Salmon’s ‘Conics,’ page 343, or ‘Bicircular Quartics,’ art. 127. 
47. If D be the spherical distance between the poles of the planes L, M, we have 
1 — 11 = 1 — 
cos D 
cos g 1 cos g" 
Hence if 1 — R=0, cosI)= cos q' cos q", or the triangle is right-angled which is formed 
by D, q', q", that is the circles S^— L, S J — M cut orthogonally (compare art. 9). 
48. The two factors 1 — R±\/ (1 — S')(l — S") of the invariant 
are plainly 
( 1 — R) 2 — (1 — S')(l — S") 
tan q' tan q" sin C, | 
tan q 1 tan q" cos \ C, J 
(51) 
wffiere C is the angle of intersection of the circles S= — L, S J — M; and these are respec- 
tively the sine squared of half the direct common tangent, and the sine squared of half 
the transverse common tangent of the two circles. We have therefore, from the exten- 
sion of Ptolemy’s theorem in my memoir “ On the Equations of Circles,” this further 
extension to conics inscribed in the same conic, namely, the condition that four conics, 
S 1 — L, S^— M, S 1 — N, S^— P, should be all touched by a fifth conic of the same form is 
V’(12)(34)±^/(ISpi)+ v /(Iip5)=0, . . 
4 p 
MDCCCLXXI. 
(52) 
