58 Colorado College Studies. 



one of the two elements given in equation (9). Now if the 

 second given circle lie wholly within the first its radins /• is less 

 than R — a, that is, 



r- < (R— af. 



But the distance g between the centres of the first and third 

 circles, as given in (9) may be factored as follows: 



'^ ~ {R — ar ' (R+aY ' 



Of these factors, the first has just been shown to be less than 

 unity; and the second is so likewise, since 



( R — af = i?- — 2nR + cr > 0. 

 Add . 4.aR = 4:aR 



( i? + « )- = i?- + 2aR + a- > 4oi?. 



We may here assume a and R are taken in their positive 

 values; hence, on the same supposition with regard to g, (which 

 must be positive if a and R are so), 



g < R. 



Hence the centre of the third circle is within the first; there- 

 fore the third circle either lies wholly within the first or inter- 

 sects it; but the latter alternative is not admissable, since, the 

 three circles having a common radical axis, if two intersect, the 

 other must pass through the points of intersection, which would 

 be contrary to the hypothesis; hence the third circle lies wholly 

 within the first. 



It will accordingly be assumed in future that the circle whose 

 radius is R is exterior to all other circles of the diagram which 

 have a common radical axis with it: each of the latter forming a 

 member of an infinite series of circles, the maximum of which 

 coincides with the circle jc' + y' — R'. and each succeeding circle 

 lies wholly within the preceding one, until the lower limit is 

 reached in the point, or circle of zero radius. This point, 

 marked G in Fig. 2, is there determined as the intersection of 

 the arc RTG Avith the axis CD; and, by the application of the 



