INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 251 



11.42. Coamplitude Formulas, with v = K, 



sn (K - u) = -7 = sn (K + u) 



sn u 



r 

 dn ( - ) = ^ = dn (K + u) 



in (K-u) = 



- - 

 tnu K'tnu 



11.43. Legendre's Addition Formula for his E. I. II, 



Ecf) = fk<t>'d(j> = ydn 2 wdw, </> = ydnw-dw = amw. 



</> + Ei/' E(T = /c 2 sin <j> sin i/' sin (r,\l/ = am v, (7 = am (v + u) 

 or, in Jacobi's notation, 



zn w + zn t> - zn ( + v) = /c 2 sn u sn z> sn (v + w), 



the secular part cancelling. 



Another form of the Addition Theorem for Legendre's E. I. II, 



77/3 Z7 / - 2 K 2 sin \// cos \I/ A\// sin 2 <j> Q , 



EG - Ed - 2E\f/ = - r 9 I 9 / ^> ^ = am (z; - w) 

 i - K 2 sin 2 4> sin 2 \f/ 



or, in Jacobi's notation, 



/ N / x 2 K 2 sn i) en i) dn v sn 2 u 



zn (v + u) + zn (z> - w) - 2 zn w = -- 5 = -- -- 



i K 2 sn 2 w sn 2 w 



11.5. The Elliptic Integral of the Third Kind (E. I. Ill) is given by the next 

 integration with respect to u, and introduces Jacobi's Theta Function, 6w, 

 denned by, 



d log Gw 



- = Zu = zn u 

 du 



0W r 



-~- = exp. I zn w^w. 

 Go J 



Integrating then with respect to u, 



. / N -, , C 2K Z snv cnv dnv sn 2 u , 



log B (t> + u) - log 6 (v - u) 2U zn v = I - 5 5 -- = - du t 



J i - K 2 sn 2 u sn 2 v 



and this integral is Jacobi's standard form of the E. I. Ill, and is denoted by 

 2 II (u, v) ; thus, 



TT / x C K 2 sn v en v dn v sn 2 u 7 , , 6 (v u) 



II (u, v) = I du = u zn v + i log TT-T \ 



J i - K 2 sn 2 it sn 2 D B G (u + ) 



Jacobi's Eta Function, Hz;, is denned by 



Hz; 



-^ = v K sn v, 



and then 



d log Hi) en v dn i> , . , , 



- = - - + zn v, denoted by zs v; 



av sn v 



