and Loci of Apollonius, &^c. 53 



the straight line RD is known in position; and .'. the point 

 D where it cuts circle A is known. 



Similarly, by drawing perpendiculars from E on the radical 

 axis RM and on the radical axis to the circles B and C, it can 

 be shown that RE through R is known in position, and .'. E 

 is known. 



And in like manner we can find the point F on circle C. 

 Or the circle DEF can be easily found from any of the points 

 D, E, F, of contact, since the lines joining these points pass 

 through known centres of similitude, &c. 



The composition may be easily made. 



NOTES. 



The ratio which DN has to PM, is as has been indicated, 



jjjj AB ■[(AC2— (rad A ~ rad C)2 1 



^^^ AC { (AB2— (rad A + rad B)'^ j 



And, in order to show that this holds good for all values of 

 the radii C and B from zero to infinity inclusive, let c and b 

 be the points in which AC and AB cut the circles C and B, 

 and let a be that in which either of these lines cuts circle A. 

 We have AC ^ Ac + cC, rad A = Aa, rad C = Cc, rad 

 B = B6; and .'.we can put the ratio under the form 



jjjj AB [ (Ac^ + 2 Ac.cC— Aa^ ± 2 Aa.cC } 

 ^^ AC { (M^ + 2 A6,6B— Ao^ + 2 Aa.6B^ j 



which evidently holds good for all values of the radii B and C. 

 The above method of solution requii'es one circle (as A) to 

 be finite. When circles C and B are infinitely smalls 



DN _ AB Ac2_Aa2 _ AB AC2— Aa2 

 DM ~ Ac'a62— A«2 = AC AB2— Ao2 



When circles C and B are infinitely greats 



Pj^^ A^±Aa 

 DM A6+Aa* 



WTien c is infinitely great and b = zero 



DN 2.AB(Ac+^Aa) 

 DM~ 



AB2 — Aa2 



When c = zero, and b infinitely great 



DN 



AC^i— Aa2 



Dai 2.AC (A6 + Aa) 



From Gergonne's solution, and the theorem on which the 

 above solution depends, it is evident we have expressions for 



