APPENDIX 



In the Legendre-Jacobi-Abel system for computing elliptic functions, the 

 following formulas, and tables, will be found useful. The formulas were 

 compiled from the following list of reference texts : 



Smithsonian Mathematical Formulae, Edwin P. Adams, Ph.D. 

 Theory of Elliptic Functions, Harris Hancock, Ph.D. 

 Elliptic Functions, Alfred Cordew Dixon, M.A. 

 Functions of Complex Variable, James Pierpont, L.L.D. 

 Advanced Calculus, Edwin Bidwell Wilson, Ph.D. 



Our use of the notations snh (u, k), en h (m, k), dnh (u, k) in the formulas 

 below may be strange to the reader, but they seem to be a natural extension 

 of the hyperbolic notation to the sn (u, k), en (u, k), dn (u, k) elliptic func- 

 tions. We have used them for years ; we hope they will be acceptable to other 

 computers. 



The basic number q may be computed as follows: given the modular 

 angle 6, the number / is computed from 



, , 1 - V cos 61 , 



/ = 4 • , , whence 



^ 1 + Vcos^ 



^=/+2/5 + i5/9 + l50P + 1707F + 20910P+268616P + . . . . 

 For 16-place accuracy this becomes difficult, if not impossible, for 6>70° ; 

 for these we may use the relation 



In q-ln g' = 7r^ 

 The Jacobi functions : 



e^(v,q)=2qi(sm ifv-q^ sin 37rt/+g« sin STrt/-^'^ sin 7-nv+. . . .) 

 e^lv, q) =2qilcos -rv+q"^ cos 37rv + q^ cos SiTV+q^^ cos 7ttv + . . . .) 

 Oz{v. q) = l+2q cos 27rt/+25* cos 47rt/+2g« cos 67rt;+2g" cos 87rt;+. . • • 

 ej^v, q)=l-2q cos 27n; + 2q* cos 4^-2q'> cos 67rt/+2(7" cos 27rt/-. . . . 



whence 



do(q)=2qHl+q' + q' + q'' + q'' + q'° + q*' + - ■ • •) 

 e,{q) = l-{-2q + 2q' + 2q' + 2q'' + 2q" + 2q'' + . . . . 

 eo(q) = l-2q + 2q*-2q' + 2q''-2q'' + 2q''-. ■ • • 



To compute K, K' we have 



K' 



-^ =dsiq), and, since q = e ^ , it follows that K'= In q. 



To compute E, E' we have 



^ £'_ 2,r= r q-Aq* + 9q'-\6q^' + 2Sq''-. . . . "1 

 ^~~K-~K^\l-2q + 2q*-2q' + 2q''-2q'' + . . . .J 



359 



