and Loci of Apollonius, i^c. 61 



same line AN = xVN ' so that ^ and ^, have like signs ; 

 find O in AL so that OC = ON ; then from the point O as 

 centre and the point D, when NO cuts circle A (so that ND 

 = N'D) as distance^ describe a circle : this circle will be as 

 required. 



NOTES. 



This method of solution is not intelligibly applicable to 

 those states of the data in Avhich any of the given circles is 

 supposed infinitely great, or replaced by a straight line ; but 

 for the other cases it is thoroughly comj^lete and deserves 

 attention. 



Indeed, T may remark that there are many very good solu- 

 tions applying only to the cases in vrhich none of the given 

 circles A, B, C, is infinite, or Avlien two are infinite. Yet it 

 must not be lost mind of, by those who would succeed, that 

 the solutions to general questions are often arrived at from 

 considering them under some particular states of the data, and 

 divining what modifications are necessary so as to make the 

 solutions which may be arrived at applicable to the more 

 general cases. 



It may also be remarked that a theorem evolved in the 

 above solution is directly applicable in a solution to the prin- 

 cipal case of the 'Inclinations' of Apollonius. 



APOLLONIUS ORIGINAL SOLUTION. 



(See Plate.) 



To describe a circle to touch three given circles A, B, and 

 C. 



ANALYSIS. 



Let D, E, and F be the points of contact of the required 

 circle with the given circles A, B, and C. 



Then, DE passes thi'ough O a centre of similitude of the 

 circles A and B ; DF passes through P a centre of similitude 

 of the cii'cles A and C; and EF passes through Q a centre of 

 similitude of the circles B and C ; and the points O, P, and 

 Q are in a straight line. 



Let G and H be the other points in which DE and DF cut 

 the circle A. 



Through H draw a parallel to OPQ; and through the other 

 point I in which it cuts the circle A, draw DI to cut OPQ 

 in M. 



