402 



8. Since the discriminant of the equation (10) does not vanish, it 

 follows that it is not the product of two simple factors of the form 



\S + fiS f + vS" = 0 ; 



Hence the equation of a circle (2) touching three circles, S, S' } S", 

 cannot be expressed in the form \S + /iS f + v8' f = 0. 



9. The result of Art. 8 may he proved independently, as follows, 

 and we can thence infer, conversely, that the discriminant of equation 

 (10) ought not to vanish :— 



For, if possible, let the equation of a circle (2) touching three 

 circles 8, S', be of the form XS + pS' + vS" E 2. 

 Hence, 



"Now, since 2 touches S", the circle XS + pS' = 0, which is coaxal 

 with 8 and S', also touches S ,r at its point of contact with 2 ; but we 

 have seen (Art. 6) that the circle coaxal with S and S', which passes 

 through the point of contact of 2 with S", cuts S", instead of touching 

 it. Hence the equation of a circle touching 8, S' f 8", cannot be of the 

 form \S + pS' + vS" - 0. — q. e. d. 



This conclusion accords with the fact that three circles, 8, 8', 8", 

 being given, the form \8 + fi8' + vS" = 0 is not sufficiently general to 

 express the equation of any fourth circle. For the equation of any 

 circle contains three independent constants, while \8 + jiS' + v8" = 0 

 contains but two, viz., the ratios \ : v and ju : v. 



10. The equations (11), (12), (13), of Art. 6, being all of the 

 form R % = LM, hence the pair of circles */lS + */ mS' + V n8" also 

 touches the circles 



l8-2m8' -2nS"=0; 

 mS> - 2n8 ' - 218 = 0 ; 

 n8" - 218 - 2mS' = 0. 



Again, the circle IS + mS' + nS" evidently passes through the inter- 

 sections of the pairs of circles 



8 and IS - 2mS' - 2nS" ; 

 £'and mS'-2nS"-2lS\ 

 £"and n&' - 2iS - 2m& ; 



