and Loci of Apollonius, S^-c. 57 



NOTES. 



Since there are two points P and two points Q^ there are 

 four lines IR, and as there are four corresponding circles CL 

 and two chords CL in each, it is obvious there are eight 

 answerable circles O real or unreal, in pairs, according as the 

 circles LCM are greater and less than the corresponding 

 circles m'n'. 



It is evident that when the circles A and C are infinitely- 

 small, and B finite, then may the point Q, have any position 

 whatever in the straight line through A and B (because the 

 infinitely small circles may have any ratio whatever just 

 according as we suppose two circles to have any finite constant 

 ratio during their diminution to the infinitely small state.) 



This solution does not readily apply to the case in which 

 two of the given circles are supposed infinitely great, or 

 replaced by straight lines ; but the following is an analysis of 

 a solution which will embrace all the cases in which we sup- 

 pose the circle A of finite magnitude. 



Since the rectangles PD.PE and QD.QF are known in 

 signs and magnitudes, it follows (from one of a class of po- 

 risms to be included in subsequent developments) that Ave 

 know the two points o', o' , of real or imaginary intersection 

 of all circles having their centres in PQ- and respective 

 radii equal the tangents from them to circle DEF. And 

 since the circle A and straight Ime PQ are known,, we 

 know the two points a! , a! , of real or imaginary intersection 

 of all circles having their centres in PQ and respective radii 

 equal to the tangents from them to circle A. And it is 

 evident we know the cii-cle through the foiu" points o'o'a'a', 

 and that its centre B is in PQ ; moreover it is evident the cir- 

 cumference of this circle R passes through D ; and . • . D on 

 circle A is known, and hence AD and the point O in which it 

 cuts the straight line tlirough o'o' ; and .'. the circle DEF 

 is known 



EIGHTH SOLUTION. 



(Sec Plate.) 



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



ANALYSIS. 



Let O be the required centre, and let D, E, and F be the 

 points of contact with the given circles A, B, and C. 



