400 



In like manner, 



and 



1, 



{It -r){H- r') ' 



making these substitutions, the equation 



cos \A a + cos \B + cos \C 7 = 0 



becomes transformed into 



>/lS + </inS' + x/nS" = 0. 



The equations of the other pairs of circles may be similarly derived 

 from the equations of the exscribed circles. — q. e. d. 



6. The equation 



</W + y/mS' + */nS" = 0, 

 when cleared of radicals, becomes 



PS 2 + ?n 2 S' 2 + n 2 S" 2 - 2lmSS' - 2mnS'S" - 2nlS"S = 0 (10) 

 Now, since this may be written in either of the equivalent forms 



(IS - mSJ + nS" {nS" - 21$ - 2mS>) 

 (mS'- nS'J + IS {IS- 2mS' - 2nS") 

 {nS" - IS) 2 + mS' {mS' - 2n$" - 21S) 



it follows that the pairs of circles 



yiS + */mS' + = 0 



touch S at the points S = 0, mS' - nS" = 0 



S'=0, nS"-lS=0; 

 $"=0, lS-mS' = 0. 



(11) 

 (12) 

 (13) 



S f 



s» 



Hence, we have the following method of constructing the points of 

 contact on the circles S, S\ S", with a pair of their tangential circles: — 



