405 



And we see that from the relation between the system of eight circles, 



«» A 7> «' s ft 7'» g '; 



and six circles, 



(a), (b), («), (d), (e), (/), 



every circle of the former system is touched by three of the latter, and 

 every circle of the latter by four of the former. — q. e. d. 



13. A very simple geometrical demonstration can be given of this 

 part of Dr. Hart's theorem; — in fact, it is inferred at once from the 

 following principle, which occurred to me some time since : — 



If two circles, P' } Q', be the inverse of two other circles, P, Q, 

 with respect to the same circle, X, the four circles, P, Q, P f , Q f } have 

 four common tangential circles. 



This is evident. 



14. Proof for the system of eight circles :— Let S, S', S" (fig. 4), 

 be the three given circles, any or all of which may be right lines, and 

 a, p, 7, S', four circles described touching them similar to the exscribed 

 and inscribed circles of a plane triangle, I say, a, (3, 7, d' } are all touched 

 by a fourth circle, besides the three circles S t S,' 8". 



Demonstration. — Let the direct common tangent to a pair of circles, 

 a and p, for instance, be denoted by the notation ap, and the trans- 

 verse commmon tangent by a/3 with an understroke ; then we have, 

 attending only to the magnitudes of the rectangles, by Art. 1, since S 

 is touched by p, 7, 8', on one side, and by a on the other, 



o(3 • 75' 4- ao - £7 - 07 * fid' = 0 ; g (a) 



in like manner, 



a7 ' fib' + 70 ' afi- py ' a& = 0 ; (b) 



and 



ap ' 7^ + pB' • a7 - £7 • ad' = 0. (c) 



Hence, by adding equations (a) and (b), and subtracting equation (c), 

 we get 



a$ ' £7 + 76' • aP - Ph' • a7 = 0. 



Hence the circles a, p, 7, are all touched by a fourth circle having 

 a, p, 7, on one side, and on the other ; hence we have the following 

 system of eight circles : — 



R. I, A. PEOC. — VOL. IX. 3 H 



