The Elliptic Functions Defined. 69 



cal axis of the system ; and accordingly, when the scale of the 

 diagram has been fixed by choosing a value of B, there is but 

 one independent constant which is required to determine its 

 proportions that, namely, which fixes the position of the radi- 

 cal axis. It has been found most convenient to define this posi- 

 tion by means of the ratio of the diameter of the circle of the 

 amplitude to the distance of its origin from the radical axis. 



AF 

 This ratio, which is -— p i^^ ^^^ figure, is denoted by k'; and the 



quantity k is called the modulus. 



To turn from the fixed to the variable elements of the figure 

 it is manifest that whatever is dependent on the magnitude 

 of (f may also be regarded as depending on that of u. Thus the 

 trigonometric functions of <p are functions of u as well. In par- 

 ticular, the sine and cosine of c, regarded as functions of the 

 argument, «, are denominated respectively sn u and en u, (read 

 " sine-amplitude H " and "cosine-amplitude it"). If P be the 

 termination of the arc 2cr, (Fig. 5) these two functions are evi- 

 dently geometrically represented, — at least so far as regards 

 their absolute value, — by the ratios which the distances of P 

 from the two extremities of the horizontal diameter FA bear to 

 the length of this diameter. If now we also draw that circle 

 of the interior system which is tangent to the chord AP, the 

 right triangles FPA and cTA ara similar; and if r and a, as 

 before, denote cT and cC, we easily find, to determine the length 

 of AT, (which we will call t,) the equation 



f= {R + af—r: (M 



Whence 



AP AT 1 



'" " = ^ = tr = R-^i V{R + ar- r^ ; (15 

 and 



FP cT r ._ 



en u = ^ , = — j- = ^5"- ( Id 



FA cA R + a ^ 



To these two functions is added a third, called the delta of 

 <P or the delta-amplitude of n, — written dn u, — and defined as 



