24 6 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



or ; = sn u dn u 



du 



d\(h ddnu 



= K 2 sin d> cos <b or -= = K 2 sn u en u 

 du du 



7T 



11.11. The complete integral over the quadrant, o<4><-,o<x <i, defines 

 the (quarter) period, K, 



making 



sn K = i 



en K = o 

 K' is the comodulus to K, K 2 + K' Z = i, and the coperiod, K', is, 



j - K' Z sin 2 <) 



11.12. 



sn 2 u + en 2 w = i 



en 2 u + K 2 sn 2 u = i 

 dn 2 u - K 2 en 2 u = /c' 2 . 

 sn o = o, en o = &a,l #xi o = i, 



11.13. Legendre has calculated for every degree of 6, the modular angle, 

 K = sin 6, the value of F</> for every degree in the quadrant of the amplitude </>, 

 and tabulated them in his Table IX, Fonctions elliptiques, t. II, 90 X 90 = 8100 

 entries. 



But in this new arrangement of the Table, we take u = F<t> as the independent 

 variable of equal steps, and divide it into 90 degrees of a quadrant K, putting 



u = eK = -^o K, r = 90^. 



90 



As in the ordinary trigonometrical tables, the degrees of r run down the left of 

 the page from o to 45, and rise up again on the right from 45 to 90. Then 

 columns II, III, X, XI are the equivalent of Legendre's Table of F<p and </>, 

 but rearranged so that Fffr proceeds by equal increments i in r, and the incre- 

 ments in 4> are unequal, whereas Legendre took equal increments of </> giving 

 unequal increments in u =' F<. 



The reason of this rearrangement was the great advance made in elliptic 

 function theory when Abel pointed out that F(f> was of the nature of an inverse 

 function, as it would be in a degenerate circular integral with zero modular 

 angle. On Abel's recommendation, the notation is reversed, and <f> is to be 



