417 



37. Again, taking any pair of the circles S, S\ S", the four points 

 of contact on it with any pair of the tangential circles (50)-(53) are 

 concyclic ; the equations of the circles passing through these concyclic 

 points are, if we denote the determinants (50)-(53) by the notation 



01, 02, 03, 04 5 



0; (78) 



0 ; (79) 



0 ; (80) 



0. (81) 



#1 

 dS 



= 0; 



dfa 

 dS' 



= 0; 



dS" 



d<?> 2 

 dS 



= 0; 



d(£> 2 



w 



= 0; 



dS" 



d(p 3 

 dS 



= 0; 



#3 



dS' 



= 0; 



d(p 3 

 dS" 



^04 

 dS 



= 0; 



dfa 

 dS' 



= 0; 



d(pi 

 dS" 



IY. 



Equations or the Conics in Paies haying double Contact with a 

 giyen Conic which touch three othees also haying double 

 Contact with the same giyen Conic. 



38. The equations of the circles on the surface of a sphere that we 

 have employed hitherto denote hut one of the intersections of a cone 

 with a sphere whose centre is at the vertex of the cone ; thus, if a, ft, 7, 

 be three such circles, then, taking account of the complete intersections 

 of the sphere with the cones, it is evident we get three other circles, 

 which we may denote by d, ft', 7'; thus we have 



8 circles touching a, ft, 7 ; 



8 „ a, ft, 7 ; 



8 „ «, P, 7; 



8 „ o, ft 7 ; 



hence we have 32 circles in all. 



The equation S- L % = 0, of a small circle on the surface of the sphere, 

 given in Dr. Salmon's " Geometry of Three Dimensions," is the com- 

 plete intersection of the sphere with the cone ; and it is easy to see that 

 its factors S 1 * - L = 0 and S i + L = 0 are the separate circles which make 

 up the complete intersection ; in fact, taking the equation of any small 

 circle on the sphere, 



Cos r - { cos ft cos p + sin n sin p cos (0 - m) J = 0, 



