INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 257 



11.9. In dealing practically with an E. I. Ill it is advisable to study it first 

 in the algebraical form of Weierstrass, 



(s - ff)VS 



where S = 4-5 - Si-s - s 2 -s - 5 3 , 2 the same function of ff, and begin by ex- 

 amining the sequence of the quantities s, cr, 5i, 5 2 , 5 3 

 Then in the region 



5>5i>5 2 >(T>53, 



put 



5 - 5 3 = ^- 3 . <r -*.'-(*- ) sn, >? = f f 



sn 2 u 5i 53 



sn 



2 - 5 3 ) sn v en dn v, making 



AV2 ^5_ /"^sn 

 J s - <T ^/S J i - K sn w sn 

 But in the region, 



i 



, _ , 3 = ( S , - 53) sn 2 ^, cr - 5 3 = 



fr _ 5es !^A 3 ( I . 



making, 



en z> dn y , 



v2_^= r^ 



1 V5 J i- 



In a dynamical application the sequence is usually 



S>5i>CT>52>5>53 



or 



5>5i>5 2 >5>5 3 >0-, 



making S negative, and the E. I. Ill is then called circular; the parameter v 

 is then imaginary, and the expression by the Theta function is illusory. 



The complete E. I. Ill, however, was shown by Legendre to be tractab 

 and falls into four classes, lettered (I 1 ) (m'), p. 138, (*0, (*') PP- J 33> *34 ( 

 tions elliptiques, I). 



