INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 255 



is negative again in the last interval, and the modulus /c'. 



S 3 >S > CO 



cn-ilA- 2 -^ 3 = dnV*-*'*- 



" S 2 - S \ S , - Ss-So- 



s 



s 2 -s 



11.8. For the notation of the E. I. II and the various reductions, take the 

 treatment given in the Trans. Am. Math. Soc., 1907, vol. 8, p. 450. The Jacobian 

 Zeta Function and the Er, Gr of the Tables, are denned by the standard integral 



Sa V S 



or, 



F 



Js2 



2/ 



P 

 Jo 



dn 2 (/') - d(fK') = E am/*' = fH' 



where zn is Jacobi's Zeta Function, and #, H r the complete E. I. II to modulus 

 AC, K', denned by, 



H = 



, /c) 



(eK)-d(eK) 



zn u 



, K') </(/> = J^dn 2 (fK'}'d(fK'). 

 The function zn u is derived by logarithmic differentiation of 0w, 



/7 Irvrr ft')^ 



-, or concisely, 



6w = exp. Jznu-du, 



and a function zs w is derived similarly from 



d log Hu 



zs u 



log Qu d log sn M 

 du du 



= znu -\ 



en w dn u 



sn u 



For the incomplete E. I. II in the regions, 



and 



i 3 

 sn 2 eK = J - or 



s 



