403 



and is therefore coaxal with each pair. Hence the three lines joining 

 the centres of these pairs of circles are concurrent. Hence we have the 

 following theorem : — 



The pair of circles v / 'lS + \/ mS' + \/ nS" = 0 touching three circles, 

 S, S', S", also touches the three other circles 



IS -2mS' - 2nS"=0; 

 mS'-2nS"- 2lS = 0; 

 nS"-2lS-2mS'=0; 



and the lines joining the centres of these circles to the centres of 

 S, S', S", respectively, concur to the centre of IS + mS' + nS". 



11. Since the equation (10) may be written in either ofthefol- 

 - lowing equivalent forms, 



{IS + mS f - nS'J = AlmSS' ; (18) 

 (mS f + nS" - IS) 2 = 4m?i'SS" ; (19) 

 (nS" + IS - mSJ = 4nlS"S ; (20) 



we have the following theorem : — 



The equations of the circles passing through the points of contact of 

 the pair of circles IS + \/ mS' + \/ nS" = 0 with 



S, S f is IS+mS' -nS' r =0; (21) 

 S', S" „ mS' + nS" - IS=0; (22) 

 S" S „ nS"+ IS -Sm"=0. (23) 



12. We shall conclude this part of the subject of this Paper by 

 applying our principles to prove Dr. Hart's celebrated extension of 

 Peuerbach's theorem : — 



" Taking any three of the eight circles which touch three others, a 

 circle can be described to touch these three, and to touch a fourth circle 

 of the eight touching circles." 



8 ■ 7 * 6 



Now, since the combinations in threes of eight things is = 56, 



1*2*3 



this theorem makes it necessary that we should have fourteen circles, 

 which we divide into two systems of circles — one system containing 

 eight circles, and the other containing six. 



Since the eight circles which touch three given circles are, four of 

 them, the inverse of the other four, with respect to the circle which 

 cuts the three given circles orthogonally, they may be denoted in pairs 

 thus : — 



aa'; f3fV; T{ > ; ^ 



