DE. J. CASEY ON CYCLIDES AND SPHEEO-QTTAKTICS. 
615 
56. The condition that four small circles 
S*-L, S*-M, S*-N, S*-P 
should be coorthogonal is easily seen to he the determinant 
cosg' , 
cos a! , 
cos j3 r , 
cosy' , 
cos f , 
cos a," , 
cos (3" , 
cos y" , 
cos f , 
cos a'" , 
cos ($"' , 
cosy'", 
cos f", 
cos a,"", 
cos / 3 
cos y"", 
Now, if we form the equations of J', J", J'" (see art. 52, Cor.), and using transformations 
similar to those of art. 54, we see that the four J’s (J, J', J", J 1 ") fulfil the condition of 
being coorthogonal. Hence we have the following theorem : — The four circles are 
coorthogonal which are orthogonal to the four triads of circles, S^dbL, S*±M, S^dbN. 
57. The plane of the great circle coaxal with J and the small circle S J — L=0 is the 
polar plane of the point whose coordinates are 5, ih (see art. 53) with respect to the 
cone S— L 2 = 0 ; and this is easily found to be the determinant 
L , 
X , 
y 
z 
COS g' , 
cos a' , 
cos/3' , 
cos y' 
cos f , 
cos a !' , 
cos (3 " , 
cos y" , 
cos f, 
cos a'", 
cos (3"', 
cos y"'. 
. (61) 
Compare art. 13. 
58. From the condition (art. 56) that four small circles on the sphere should be 
coorthogonal, it is easily inferred that the poles with respect to the sphere of the planes 
of these circles are complanar. Hence the planes of these small circles pass through a 
common point. Conversely, any four small circles on the sphere are coorthogonal whose 
planes pass through a common point. 
Cor. The common point through which the planes of four coorthogonal circles pass is 
the pole with respect to the sphere of their common Jacobian. 
59. The orthogonal circle J, as will appear evident from the form of its equation (see 
art. 55, equation 59), has double contact with each of the four sphero-conics : 
cos 2 \ A, 
cos 2 | B, 
cos 2 C 
x tan g' 
y tan g" 
tan g'" 
cos 2 | A 
sin 2 1 B 
sin 2 | C 
x tan g 1 
y tan g" 
s tan g 1 " 
— sin 2 1 A 
cos 2 | B 
sin 2 ! C 
x tan g 1 
r y tan g" 
£ tan g'" 
— sin 2 1 A 
sin 2 B 
cos 2 | C 
x tan g' 
y tan g" 1 
2 tan g 1 " 
. . (62) 
. . (63) 
. . (64) 
. . (65) 
