412 



27. Again, if S, S\ S", S'", be the inscribed and exscribed circles 

 of a spherical triangle, we have 



Sin J common tangent of S, S' x sin \ common tangent of S", S ,rr 



= sin 2 jjb - sin 2 \c ; 



and the other rectangles = sin 2 \e - sin 2 -Ja, and sin 2 \a - sin 2 \b, respec- 

 tively. Hence the condition holds of the circles S\ S", S'", being 

 all touched by the same circle. — q. e. d. 



28. It is evident that the three anharmonic ratios of the points of 

 contact are 



b 2 -c 2 e 2 -a 2 a 2 -b 2 

 c 1 - a 8 ' a 2 - b 2 ' b 2 - c 2 ' 



(42) 



for plane triangles ; and for spherical triangles, they are 



sin 2 \b- sin 2 J c sin 2 \c - sin 2 \a sin 2 \a - sin 2 ^b 

 sin 2 \c - sin 2 \a sin 2 \a - sin 2 sin 2 - sin 2 \c 



(43) 



29. Let P be the centre of a small circle S on the surface of a 

 sphere (fig. 6) ; 0 a fixed point also on the surface, which we shall 

 take as origin : OX a fixed great circle, corresponding to the initial line 

 in plane geometry; and let OP = n, the angle POX=m, and the co- 

 ordinates of any point Q of the circle S be p and 9, then we have from 

 the spherical triangle OPQ, r being the radius of the circle S, 



cos r= {cos n cos /> + sin n sin p cos (0- m)} = 0. (44) 



This may be taken as the equation of the small circle S ; and it is plain 

 that with this system of co-ordinates the result of substituting the co- 

 ordinates of any point Q f in the equation of a small circle S on the sur- 

 face of a sphere is equal to 



2 cos r x sin 2 \ the tangent from Qj to S. 



This may be written 



2 cos r x sm 2 \t =S 



sin 



2 cos r 



(45) 



30. If the small circle S'" of Art. 26 become a point, and if we 

 denote the 



sin \ direct common tangent of S\ S", by 



S „ m\ 



