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DE. J. CASEY ON CYCLIDES AND SPHEEO-QTTAETICS. 
7T 
S^— L, S 1 — M, S* — N be great circles, we have q'=q"=q"'=~, and I, H, K each = zero, 
but G is finite. Hence the orthogonal circle becomes the imaginary circle at infinity. 
Hence we have' the important theorem : — That the imaginary circle at infinity is the 
circle which cuts three great circles of the sphere orthogonally , and is therefore a limiting 
case of the circle cutting any three circles orthogonally . 
Cor. In the geometry of a plane the two circular points at infinity are represented by 
the circle which cuts the three sides of any triangle orthogonally, and is therefore a 
limiting case of the circle which cuts any three circles on the plane orthogonally. 
54. The following transformation of equation (55) will be useful in a subsequent 
article. Let the sides of the spherical triangle formed 
by joining the spherical centres of the three small circles 
— L, S* — M, S ir — N be denoted by ' , fi', ^ " respec- 
tively, and the direction cosines of the planes of these 
sides, or, which is the same thing, of the lines from the 
centre of the sphere U to the angle points of the supple- 
mental triangle becostf', cos V, cos c'; cos a", cosZ»", cos c"; 
cos a'", cos l)" 1 , cos d" respectively, then the equation (55) 
becomes transformed into the following : 
& 
cos a' , cos (3 r , cos f 
+ cos q’ sin ^ ( x cos a! -\-y cos V -\-z cos c' 
cos u," , cos j3" , cos y" 
= . 
+ cos q" sin ^ " (x cos a" -\-y cos h" -\-z cos c" 
cos a 1 ", cos ft" 1 , cos y 1 " 
+ cos q" 1 sin \P" \x cos a"’-\-y cos b"'-{-z cos c'". 
55. Let us seek the locus of all the point circles of the system 
Ify — L) -f- m( — M) +w(S> — N). 
The equation —X cos X-\-y cos l JjJ r z cos v denotes a point circle (see art. 50). Hence, 
comparing coefficients, we get 
cos X = l sec q 1 cos a' -j -m sec q" cos a" -J -n sec f cos a'" 
cos [Jj—1 sec q’ cos ft'ft-m sec q" cos ft"-\-n sec q"' cos / 3 
cos v =1 sec q' cos y’ -\ -m sec q" cos y" -\-n sec q'" cos y '" ; 
square and add, and we get 
1 = l 2 sec 2 g>'-j -m sec 2 f-\-n 2 sec 2 f + 2 Im sec q' sec q" sin f" 
+ 2 mn sec q" sec q'" sin ^'-\-2nl sec q'" sec q' sin 
Now, since all that we are concerned with is the mutual ratios of the multiples l , on, n, 
let us suppose l-\-m-]-n=l ; square and subtract from equation (58), and then replace 
l, on, n by x, y, z, and we have the equation of J (see art. 51) in the form 
(tan 2 q 1 , tan 2 q", tan 2 q"', — tan q' tan q" cos C, 
— tan q" tan q'" cos A, — tan q'" tan q' cos BJ/r, y, z) 2 = 0. 
Compare ‘Bicircular Quartics,’ art. 139. 
Fig. 2. 
