410 



milar equations from equations (29)-(35), inclusive ; and since from 

 any three of these equations we get eight systems of common values for 

 X, /x, v, p, we infer that the three pairs of spheres denoted by any three 

 of the equations (28)-(35) are touched by eight spheres, four of which 

 are the spheres S, S' 9 S", S'". 



23. The eight tangential equations can all be included in one 

 general formula, as follows : — 



Let the radii of the four spheres, S, S', S" S' r; , be denoted by 

 r, r', r", r f// , and the angle at which S intersects $' by the notation 

 (SS'), then we have 



2= Ar'r" cos 2 ±(S' S") ; 

 V^-Wr" sin 3 i(S'S"); 



and similar values for m, m,' &c. Hence the equation (37) becomes 

 transformed into 



Xfxrr' cos 2 1(&S") + fxvr'r" cos 2 \(S' S") + vpr'V" cos 2 \(S" S>") 

 + Xvrr" cos 2 \(SS") + tyrr'" cos 2 \(SS'") + ppr'r'" cos 2 \(S'S m \ 



This is equivalent to the equation 



J7= (Xr + fir* + vr" + pr'") 2 , (38) 



where 



U E X 2 r 2 + p?r n + v 2 r" 2 + p 2 r"' 2 

 -2\firr' cos (SS') - 2/ipr'r" cos (S /; S") - 2vpr"r"' cos (£"£'") 

 -2Xvrr" cos (SS") - 2X P rr" f cos (SS'") - 2jupr'r"' cos (S'S"'). (39) 



And the eight tangential equations are included in the formula 



U = (V ± fir' ± vr" ± pr'J (40) 



(Compare Salmon's " Geometry of Three Dimensions," Art. 219.) 



lit 



EQUATIONS OF THE CIRCLES IN PAIRS WHICH TOUCH THREE CIRCLES ON A 



SPHERE. 



24. The theorem, Art. 1, which was proved by inversion, can be 

 proved without inversion, as follows : — 



Let 0, 0', 0", 0"', be the points of contact ; A, B, C, D, the cen- 

 tres ; r, r', r 7f , r' f/ , the radii of four circles, S, S', S'\ S"' } which touch 



