418 



it is by transformation to three rectangular planes changed into 



(ar + y 2 + s 8 )* cos r = L, 



where x, y, %, are the co-ordinates of any point in the circle, and L is 

 the perpendicular from the same point on the plane of the great circle 

 whose pole is the centre of the small circle ; now, this equation is of the 

 form 



and it may be shown that the equation of its twin circle (see Salmon, 

 page 200, foot note) is of the form 



Hence the equation of the pair of circles touching three circles, a, /3, 7, 

 on the surface of a sphere may be written in either of the forms 



\ cos r * cos r' * cos r" 



y cos r \ cos r \ cos r" ^ 



(82) 



(38) 



And it will be seen that these, when cleared of radicals, give the same 

 result ; and the equations of the other pairs of circles are got from these 

 by properly accenting I, m, n. 



39. If S' S", be two small circles of the sphere (fig. 7), abed a 

 great circle passing through their centres, and J any circle cutting 

 them orthogonally in a', V , c , d' ; now, the anharmonic ratios of the 

 four points, a, b' , c\ d', are equal to the anharmonic ratios of the points 

 a, b, c } d; and two of the anharmonic ratios of the points a, b, c, d, are 



sin \ ac • sin J Id : sin ± ab ■ sin \ ed ; 

 sin J ad ■ sin -J be : sin -J ab ■ sin -J cd; 



and these are respectively equal to 



I : tan r • tan r" ; 

 I : tan r' • tan r". 



Hence we have the following theorem :— 



If any circle J cuts two small circles, S', S", on the sphere ortho- 

 gonally, two of the anharmonic ratios of the four points of section are 



I : tan r ' tan r" ; 

 V : tan r • tan r"; 



