415 



And if the circles S, S', S", become great circles, denoting them by 

 a > P) 7> we get for the equation of the inscribed circle of a spherical 

 triangle — 



cos %A^/a + cos 2 J B*/~p + cos iCS/7~=0 ; (59) 



and the equations (47)-(49) give, when similarly transformed, the 

 equations of the exscribed circles. 



34. The tangential equations (54)- (57) become, by the substitu- 

 tions of the last articles, 



cos 2 1.4 cos 2 %B cos 2 JC _ 



7 2 + r-^y + r^— = 0 J (60) 



A. sm t ft sin r v sin 



cos 2 -^ sin 2 15 sin 2 iC =()j (6l) 



Xsinr ytt sm r' 1/ sm r 

 sin 2 M cos 3 ±B sin 2 * (7 



\ sin r ytt sin / 1/ sin 



0; (62) 



sin 2 |4 sin 2 cos 2 £ C _ 



— T-^ + ^ = °* ( 63 ) 



A, sm r u sm r y sin r v 7 



These formulas are all included in the general formula 



U = (X sin r ± ft sin r' ± v sin r")* t (64) 



where 



i7E X 2 sin 2 r + ^ 2 sin 2 r' + * 3 sin 2 r" 



- 2fiv sin r f sin r" cos ^4 - 2v\ sin r" sin r cos B 

 - 2\jll sin r sin r' cos C. (65) 



35. The equations (60)-(63) denote the eight circles tangential to 

 three circles on the sphere, and each pair are touched by the pair of 

 circles 



U= {X sin r cos (B-C) + /tsinr' cos C-A) 



+ vsinr"cos(.4-B)} 2 (66) 



(See Salmon's " Geometry of Three Dimensions," Second Edition, Art. 

 253). 



