404 



then it is evident that 



</lS + */mW + \/nW' = aa! 

 -/IS + V^m'J' + VrtW^ pp' 



\/TS + \/mS' + = 



Now, the equation of the pair of circles aa', when cleared of radicals, 

 is by equation (18) 



and this being of the form LM - B 2 , the equation of any circle touch 

 ing a and af will be of the form 



[i 2 Z-2[jlR + if = 0 



(Salmon's " Conic Sections," p. 234, Fourth Edition); or, restoring th 

 values of Z, M, R, 



-fi)lS - (p- \)mS f + finS" = 0. ( 



Similarly, the equation of any circle touching the pair of circles P 

 will be of the form 



(li n -*ji')lS- (/*'- l)m'S'f pn'S" = 0. (J) 



In order that equations (a) and (J) may represent the same circle, 

 we must have 



fx m 



fi m 



/a - 1 _ n 



fi! - 1 n r 



for the system of six circles. 



Hence fi and fx' are determined, which proves the proposition ; and 



we have the following system of six circles : — 



aa'pp' are all touched by a fourth circle. (a) 



aa, 77 / )) (^) 



aaW „ (c) 



PPW „ (<*) 



ppw „ (■#] 



77^' „ (/) 



