254 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



11.7. For the E. I. I and its representation in a tabular form with 



S 2 - S 3 

 Sl ~ S3 



K/2 = fLZ__ 2 , 



Sl - S 3 



rcO,S2 



K- / 



t/a. S3 



- 83 ds 



Vs 



- s 



and utilizing the inverse notation, then in the first interval of the sequence, 



eK = 



s 3 ds 



= P 



Jsi 



Vs 



- s 3 ds 



= sn 



5 



= dn- 1 ' 



'S - 5 2 

 5 



Vs 



indicating the substitutions, 



Si- 



5 52 



5-5 2 



-5 3 '5 -5 2 



S 53 



= sin 2 </> = sn 2 eK, - - = sin 2 ^ = sn 2 (i - e)K. 



In the next interval 5 is negative, and the comodulus K' is required. 



, = C Sl 



- S 



^ 3 ds 



= sn" 



is positive again in the next interval, and the modulus is K. 



5 2 >5 >5 3 



_ e)K 



5 3 ds 



VS 

 indicating the substitutions, 



sn' 



5 

 Sl- S3 



S ~ S3 



S2 ~ 83 



sin 2 = sn 2 eK 



s = 5 2 sin 2 $ + s a cos 3 4>. 



