DE. J. CASEY ON CYCLIDES AND SPHERO-QTJARTICS. 
613 
determinant 
, 
X , 
y 
z 
COS q' , 
cos a! , 
cos/3' , 
cos y' , 
COS q" , 
cos u " , 
cos (3 " , 
cos y " , 
COS q 1 ", 
cos a" 1 , 
cos (3 
cos y'", 
= 0 . 
(54) 
Compare art 11, equation (16). 
52. The foregoing equation can be got directly as follows, my object in giving the 
above method being to show the identity of the methods of spherical geometry, and the 
method of conics given in the ‘ Bicirculars and in fact it was geometrically, that is, from 
consideration of the sphere, that I first discovered the method given in the 4 Bicirculars.’ 
Let w be the radius and x, (a, v the direction-angles of the axis of the orthogonal circle ; 
then, from the condition 1 — B = 0 (see art. 47), we get three equations, 
cos q' cos r— cos X cos a' — cos (a cos {3 r — cos v cos y' =0, 
cos q" cos or — cos X cos a" — cos (a cos [3" — cos v cos y" — 0, 
cos q 1 " cos or — cos X cos a'" — cos [a cos (3 — cos v cos y'" = 0 ; 
and the required circles give us a fourth equation, 
S 1 cos or— cosx(#) — cos (A{y)— cos v(z) = 0. 
Hence, eliminating linearly, we get the same determinant as before. 
Cor. The equations of the three other J’s are got from the equation (54) by putting- 
negative signs to the direction cosines of the axes of the circles. 
53. The equation (54) expanded is 
S* 
cos a J , cos [3 1 , cos y' , 
cos a " , cos j3" , cos y" , 
cos a" 1 , cos / 3 cos y'", 
(55) 
X 
COS (3' , COS y' , COS q' , 
y 
cos y' , cos a 1 , cos o' , z 
COS (3 " , cos y" , COS q" , 
+ 
cos y" , cos a", cos q" , + 
cos (3 cos y'", cos q'", 
cos y"', cos a!" , cos q 1 ", 
COS «' , COS (3 1 , COS , 
cos a", cos/3" , cosq", 
cos a'", cos [3 cos 
Let this be written GS J = lx H_y T Hz, and comparing it with the equation 
S J =sec r(x cos X-f-y cos^ + ^cosr), 
I 2 + H 2 + K 2 
we get 
sec'f: 
Gr 2 
Hence the coordinates of the pole of the plane of the orthogonal circle, with respect to 
the sphere U, are 
1 H K 
Gr’ G’ G ( d6 ) 
Now if this point be within the surface of the sphere U on which the circles S J — -L, 
— M, S — N are described, the orthogonal circle will be imaginary. But if the circles 
4 p 2 
