397 



" If S, S ; , S", S" f , be four circles which touch a fifth circle 2, the 

 common tangent of S S' by the common tangent of S" S fff plus the 

 common tangent of S\ S" by the common tangent of S, S" f , plus the 

 common tangent of S", $, by the common tangent of /$" S //r = 0 ; the 

 common tangent of any pair of circles being direct or transverse, ac- 

 cording as their contacts with 2 are similar or dissimilar." 



The foregoing theorem being the foundation of nearly the whole of 

 the following Paper, I shall in a subsequent article give another proof 

 of it, not derived, like the preceding, from inversion, and which, 

 slightly modified, will give the corresponding theorem respecting four 

 circles which touch a fifth circle on the surface of a sphere. 



2. If we denote the direct common tangents to $', S"; S", S ; 

 S, S'; by 



U, m*, n*, respectively, 

 and the transverse common tangents by 



V*, n't; 



and supposing the fourth circle S'" to become a point, the common 

 tangents" to S"' , S; S" f , S' ; S'", S" ; become the square roots of the 

 results of substituting the co-ordinates of the point S'" in the equations 

 of the circles S, S', S /f , respectively, and hence they are 



x/s, VS', VS". 



Hence, the co-ordinates of any point in the circle touching S S f S" must 

 satisfy the equation 



^IS + ^/mS' + ^/nS'^O; (2) 



and since this equation, cleared of radicals, is of the fourth degree, it is 

 the equation of a pair of circles, 2, 2', touching S, S' } S", as in 



fig- (!)• 



In like manner, the equations of the three other pairs of circles 

 touching S } S', S", are 



^iv/^'+y^^ (3) 



yVS + v / ^& + vV£" = 0 (4) 



yfS + VmTS + <s/W' =0 (5) 



3. The equations of the inscribed and exscribed circles of a plane 

 triangle are particular cases of the equations (2), (3), (4), (5), and may 

 be inferred from them as follows : — Let the radii of the circles S, S',S", 

 be a, b, c, and let the angles in which they intersect each other be— 



for 





S"; 



A 





S", 



S; 



B 



)> 





S; 



C 



R. I. A. PROC,— VOL. IX. & a 



