V 



326 report — 1865. 



we obtain also 



O TjT °0 +00 



2 ^=^in M (i-2 2, ") 3 =^ St(-l>"(a»+i)2* <a,>w, =_fl{.i(0) (22) 



These equations (19-22) are of great importance in the arithmetical appli- 

 cations of the theory. 



The constant a having the particular value 2K, the functions 



0o,oO*O, 0cn(*)> i.oO*O> tOi.i(*) 



are denoted by Jacobi by the symbols O^x), Q(x), H^a?), H(a?) ; we shall 



find it convenient occasionally to employ this notation. 



The elliptic functions (properly so called), sin ama?, cos ama;, A ama:, are 



defined by the equations 



1 H(a?) J K '~H..(x) . , ,Q,M /OQ . 



sinamo7=- r -f-^; cosamo;=^L_— *M; Aam.r= */ k -£~{. . (23) 



These functions are all doubly periodic, having for their periods 4K, 2iK' ; 

 4K, 4iK' ; 2K, 4iK' respectively ; introducing them into the equations (16-18), 

 we obtain 



cos 2 am#+sin 2 am x=l, \ ,^a\ 



A 2 ama; + k 2 sin 2 ama;=l,J 



d . sin ama; 



dx 

 d . cos am x 



dx 

 d . A am x 



=cos am x A am x. 



= — sin am x A am x, y (25) 



= — k" sin am x cos am x. 

 dx 



From these formulas it appears that if y=sin am x, x is one of the values of 



Cy dij 



the integral 1 , , ,', , =. All the values of that integral are re- 



8 Jo •(1-^)(1-«V) 

 presented by the formula x+4mK+2m'iK', in which m and m' represent any 

 integral numbers whatever. Since sin amK=l, K is one of the values of the 



fl fly 



y ; and it can be proved that R' is one of the 



Jo •(i-tfXl-.«Y) 

 Tup* 



values of the integral I . , , "f, ,„ • When the real part of w va- 



Jo •(i-y a )(i-<y) . . , ,, 



nishes (in which case q, K, K', (.-, k' are real and positive, and k, k' less than 

 unity), K and K' are the ordinary values of those definite integrals : i. e. the 

 values obtained by causing y to pass from the inferior to the superior limit, 

 through a series of real values. 



The well-known formulas of Addition and Subtraction which express the 

 elliptic functions of the sum or difference of two arguments in terms of the 

 elliptic functions of the arguments themselves, are easily deduced from (13). 

 But as we shall not require these formulas in the following articles, we may 

 omit them here. 



125. TJie Modulus and its Complement. — The Theory of Transformation. — 

 In the arithmetical application of the theory, the functions k and k, which are 

 respectively termed the modulus of the elliptic functions, and the complement 

 of the modulus are of primary importance. They are respectively fourth 

 powers of the quantities 



