For values of sn(w, k), cn(u,K), dn{u, k) where the modular angle 6 lies 

 between tabulated values, and where interpolation fails, direct computation 

 may be employed up to, and including, ^ = 89°, using equations (A) above. 

 Values, 15-places correct, can be thus computed without prohibitive labor. 

 That, in fact, was the way our tables were computed. 



For values of 6 between 89° and 90° we may resort to Jacobi's imaginary 

 transformation equations : 



(C) 



, . —isn(in,K') snh(z/,K') 



sn(M, K)= -~ — -—^ = ^~ — ^ 



en {iu,k') cnh (u,k') 



cn(M^ k) = — — = 



cn{iu,K) cnh(M, k') 



J . . dn(ni,fc') dnh(jf,/c') 



Qn{l{, k)=: — y-. -^ = ^^ — - — — 



cn(iM, /c) cnh(M, k') 



together with 



(D) 



cn(u, k) 

 dn(?/, k) 

 k', sn(!<, k) 

 dn(i<, k) 



' r 



dn(K-u, k) = —-^ 



an{u, K) 



Confining our attention to sn(jY, k), we may obtain from these, written out 

 for ready use 



(E) 



snh(it, k') 



sn(K — n, k) 

 cn(K — u, k) 



sn (u,k) = 



cnh(z/, k') 



_ ^aiq') r sinh-n-y — (/'- sinh 37rt/' + (/"' sinh 57rt/' — <7'^- sinh 77rt/ • • - I 

 ~ Ooiq') [cosh itv' + q'- cosh ^ttv' + q'^ cosh Sttz/ + q'^"" cosh 7ifi/ . . J 



u 

 where i/ = 2K' 



A ^ (ir \ zn{u,K) 1 zn{m,K'') _ 1 



and also sn(A — J/, k) = -; — r- = — j-. 7- — r—7-. tt = TTT r\ 



dn(H, k) cn(/», /c) dn(n<, /<) dnh((^K) 



that is 



(F) 



B-,{q') rl-2^'cosh27rz/'+2^"'cosh47rz/-2g"'cosh67rZ'' + . ."I 

 sn(A-i<,K)_ j-^ L 1 + 2^' cosh 27rt/ + 2c^' cosh \ifv' + Iq^^ cosh 67^/ + . . J 



where 1/ — i^^FT' 



361 



