401 



Describe the circle IS - mS'= 0 ; this circle, coaxal with S, S', will 

 intersect S" in two points, which will be points of contact. Again, 

 describe the circle mS' - nS" ; it will intersect S in the points of con- 

 tact. Lastly, describe the circle nS" - IS = 0, and it will intersect 8' in 

 the points of contact. 



7. Since the circle IS - mS' = 0 intersects S" = 0 in the points of 

 contact of S" with the pair of circles */ IS + V mS' + V nS" - 0, 



lS-mS'-(l-m) 8" = 0 



passes through the points of contact. 



Now, S - S" = 0 is the radical axis of the circles S, S", and S'-S"= 0 

 is the radical axis of the circles S', S". Hence, denoting the radical 

 axis of 



S' S" by A, 

 S" S „ A', 

 8 8 f „ A\ 



this equation becomes 

 hence, 



mA -IA' = 0 ; 



I m 



In like manner, the points of contact on 8 are constructed by drawing 

 the line 



A ' A" 



m n 



and the points on S' by drawing the line 



n I 



Hence we derive the following theorem : — 



The chords of contact of the three circles 8, S', 8", with their four 

 pairs of tangential circles, are given by the four systems of equations 



(14) 

 (15) 

 (16) 

 (17) 



A 

 I " 



A' _ 



m 



A" 

 n 



A 



A' 



A!' 



I ~ 



m! 



ri 



A 



A' 



A" 



T~ 



m 



n' 



A 



A' _ 



A 1 ' 



V ~ 



m' 



n ' 



