The Elliptic Functions Defined. 63 



the radical axis, for as the number of sides of the polygoli 

 increases these lines approach parallelism. It is to be noted, 

 however, that this part of the construction is merely the appli- 

 cation of the method given for finding on a given line the point 

 of contact of a tangent circle which has in common with a given 

 circle a given radical axis. A second method for solving this 

 subsidiary problem will be given later, and thus the above-noted 

 mechanical difficulty Avill be obviated. 



It seems well to notice here, — although it is aside from the 

 immediate purpose, — that the equations thus far obtained serve 

 also for the case of a triangle inscribed in one circle and cir- 

 cumscribed about another; and hence for the determination by 

 the above process of the elements of the inscribed circle for a 

 polygon of 3 X 2" sides. In Fig. 1, PJIX will be such a 

 triangle as just described, provided the circle touched by the 

 side JIX is identical with that touched by PJI and PX, that is, 

 if g and j) are equal to a and r respectively. Applying this con- 

 dition in the value of g given in equation (9) we have. 



" ~ {E' — cry ' 



Dividing through by a and extracting the square root we 

 obtain 



R' — a' = 2rE, (13 



as the relation which must subsist between the radii of an ex- 

 terior and an interior circle, and the distance between their cen- 

 tres, in order that a triangle inscribed in one may be circum- 

 scribed in the other. If R and a be regarded as known, r may 

 be found geometrically by constructing the proposition 



2R : R -\- a = R — a : r. 



When the circles have been determined, the radical axis may 

 be found either geometrically, by well-known constructions, or 

 analytically by equation (11); and thenceforward the work pro- 

 ceeds as in the foregoing paragraph. 



The form of the above equation (13), (which is ascribed to 



