The Elliptic Functions Defined. 



61 



Here there appears to be a new ambiguity. Both of these 

 roots are real, but one of them, as is obvious on inspection is 

 greater than B; that is, the circle to which it belongs has its 

 centre outside the circle x' -\- y- = R'; hence it is not a member 

 of the series of circles to which our attention is at present 

 restricted. But the other root, while of the same sign as g is 

 less than g; whence the circle corresponding to it lies betweeen 

 the circles whose radii are E and p, and so exterior to the latter. 

 We learn accordingly, that starting from a given base MX we 

 can always determine in one and only one way a triaugle PMN 

 whose vertex shall lie on the opposite side of MN from the 

 circle to which the latter is tangent, and whose sides, meeting in 

 P, shall touch a circle exterior to that touched by MN. 



We can now solve, both analytically and geometrically, the 

 following problem: 



Within a given circle to determine the position of a second 

 circle, having in common with the first a given line as radical 

 axis, and such that a polygon of 2" sides may be at once 

 inscribed in the first circle and circumscribed about the second. 



The process and the result 

 are entirely independent of the 

 position of the first vertex P of 

 the polygon, which may be as- 

 sumed arbitrarily upon the cir- 

 cumference of the given circle. 



The first polygon to be con- 

 structed is that of two sides, 

 which consists of a chord ex- 

 tending from P to some other 

 point of the circumference and 

 thence back to P. As the two 

 sides coincide, the inscribed circle is of zero radius, hence con- 

 sists of the point G, geometrically determined as in Fig. 2. The 

 second vertex of the polygon is therefore fixed at the point P' 

 where PG meets the circumference. 



^'f 3- 



