ELLIPTIC INTEGRALS 
207 
7. The Elliptic Functions. If we set 
x 
V(1 -x 2 )(l - k 2 x 2 ) ’ 
dx 
— l^x^l, 
the equation represents u as a function of x. The inverse function, 
x, regarded as a function of u, is called the sine amplitude of u and 
is written 
x = sm am u or x = sn u. 
Two other functions are defined by the equations: 
vT — x 2 = cos am u or cnw; 
Vl — k 2 x 2 = A am u or dnw 
(read: “delta amplitude of u” or “dnw.) These functions are 
known as Elliptic Functions, and any rational function of them is 
also called an elliptic function. For a brief treatment of them, cf. 
Byerly, l.c. A more extended study is found in Pierpont’s Func 
tions of a Complex Variable, Chapters X-XII. Cf. also, both for the 
elliptic integrals and the elliptic functions, Schlomilcli, l.c.