The Elliptic Functions Defined. 65 



them m-?i^''^ of the whole perimeter of the polygon (when this is 

 estimated by the }iit))ibe)', not the Jengfh of the intervening 

 sides,) however great the number of sides may be made. 



Let us now return to the consideration of the former of these 

 last two problems — viz., that relating to the polygons of 2" sides, 

 — and let us suppose that there is given, as in that and previous 

 problems, the circle of radius B, also the right line which has 

 been previously used as a radical axis; and we will define the 

 position of the latter a little more strictly than heretofore by 

 requiring that the diameter to which it is perpendicular, which 

 may be called the initial diameter, shall be placed horizontally, 

 while the radical axis itself stands on the left side of the circle. 

 This circle we will name the circle of the amplitude. To avoid 

 unnecessary repetitions of a statement already familiar, let it be 

 understood in future that whenever a polygon is mentioned as 

 inscribed in the circle of the amplitude, it is meant that the same 

 polygon is also circumscribed about one of the circles which 

 constitute what we shall henceforward call the interior system* 

 i. e., those which are within the circle of the amplitude, and 

 have with it the given line as radical axis — the particular circle 

 touched by any polygon of 2" sides being always uniquely de- 

 termined to each value of n, by the method already explained. 

 Let us now suppose another circle to be drawn, exterior to the 

 circle of the amplitude, and hence, of course, to the whole in- 

 terior system, and not having with them the radical axis which 

 they have in common, but being instead, concentric with the 

 circle of the amplitude. We will call this new circle the circle 

 of the argument. Its radius might, for our present purpose, be 

 of any convenient length. If, after defining the elliptic func- 

 tions, we pursue the consideration of them sufiiciently far, we 

 may discover a reason for choosing a particular length for this 

 radius, but we will at present be content to regard it as arbitrar- 

 ily taken of a length whose ratio to that of R we will denote by 

 Sir : -. The right hand extremity of the initial diameter, in 



