The Elliptic Functiom Defined. 73 



ition of the fuuction (hi k, which may now be defined to be the 

 ratio which the distance of the termination of the arc 2c from 

 the fixed point G bears to the fixed distance GA. 



We have hitherto regarded the polygons which have been 

 inscribed in the circles of the argument and amplitude, as figures 

 of the same character as the polygons of Euclid's Elements; — 

 such, that is, that in passing along the sides from one vertex to 

 the same vertex again, only one circuit is made about the center 

 of the circumscribed circle. But this restriction is entirely 

 unnecessary, and the geometrical constructions as well as the 

 analysis can easily be applied to the case of polygons whose 

 sides make any desired number of circuits. Suppose, e. g., 

 that in the construction of Fig. 3., we had chosen to regard the 

 arc extending from P to P again as consisting of 6- instead of 

 2-. The polygon of two sides, PP', would have'been constructed 

 as before, but the distance on the circumference from P to P' 

 requires a whole revolution and a half. So [also the quadri- 

 lateral presents the same appearance as before,^but the order of 

 the vertices is PP/ P' Pj. But coming"' to^the octagon, we use 

 the chord FH' instead of AH' as determining the tangent circle, 

 and place the second vertex of one figure](the^first being at P), 

 on the arc P^P'. Thence forward we proceed as before. In a 

 similar manner we may form polygons which make^five circuits, 

 or any other number. In fact, if P be^a'point on the circle of 

 the amplitude, the extremity of the mth side (counting from A) 

 of a polygon of 2 " sides, then AP will be'the side of a polygon 

 of 2" sides, all of which touch one circle of the interior system, 

 and the last of which at the close of m circuits of^the circumfer- 

 ence, returns to A. And on the same principle by which we 

 defined to every point on the circle of the argument its corres- 

 ponding point on the circle of the amplitude, we may regard the 

 chords of any two corresponding arcs, 2u and 2cr, measured from 

 the origins of the two circles, as the initial sides of inscribed 



polygons. 

 6 



