COIL WITH FOUR CIRCLES. 109 



Taking into account the ratios 



and putting 



_r_ ,_r_ 



m ~a' "V 



these equations may be written 



pnP (2i; 4 - 2%m* + 8) + m' 2 (zim'* - 2%m' 2 + 8) = 0. 



Subtracting here the first equation from the second, we may re- 

 place the latter by the simpler equation 



pm^ (7#z 2 - 6) + m' (>jm' 2 - 6) = 0. 

 We get from this 



*K' 2 m'* m' 2 - 6 



~~ 



- 6 ' 



which gives 



~ 



32 



We may observe that m and iri represent the sines of the angles 

 a and a 1 , under which the two radii r and r' are seen from the 

 centre ; their squares must always be between zero and unity ; from 

 this follows 



.-6 



rfiere 



-4 



49 



