16 Ellery Williams Davis 



The definite integrals 



f^= and f /2 <* 



o Vi—Psin^ •/ ]/!_ k' l sin*$ 



are called A' and A". Elliptic-function theory shows us that 



sn\w+K)=xd?w, sn\zv-\-iK')r=csc 1 Bns i w, 

 s?i\w+K+iK') =csc 2 0dc 2 w, 



with similar equations for the other forms. If, now, /(«/) denotes 

 any one of the 24 forms, 



/(te/).(a^)(yS)=/( 7 ^ +/A - )) e.g. / 3 (^+ 2 -A")=/l(^), 

 /"(^).(ay)(/3S)=/(^+A'+/A"), e.g. / 2 ( a ,+A'-f^) == / 3 ( a ,). 



When we wish to consider the functions dependent not only upon 

 w but also k we write f(w, *). Theory shows us that we can 

 change to a reciprocal modulus by 



PstPQw, k)~sn$kw, ~ \ 



or sm 2 0s?i-(w, sinO)— sri* {w sinB } csce) 



he - s?i 2 (w, sw6)( y 8)=sn 2 { wsi?i6, csce), 



or more generally f x (w, sine) {yh)=^{w sine, csc6), 

 and A( w > sine )( y 8)=/ z (zc> sine, csc6). 



But also cd \ *», ^]=dc\w, k)=cd\w, k) (yB), 



whence f 2 (wsine, &$)=&(&, $m0)( y S); 



while i ^ [w^]=£<fc»(w, ^-[^(e*. *)] (yS), 

 so that / 4 0, sine)( y h)=f i {zvsine, csce). 



Thus the subscripts can be dropped and we can write 

 f(w, sine) (yh)=f(w sine, csce), 



or ' f(w,k)(y8)=f[kv;lj 



246 



