INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 253 



The Jacobian multiplication relations of his theta functions can then be 

 rewritten 



D(r + s)D(r - s) = D 2 rD 2 s - tan 2 OA*rA*s, 



A(r + s)A(r - s) = A*rD*s - D 2 rA 2 s, 

 B(r + s)B(r - s) = B 2 rB 2 s - A 2 rA 2 s. 



But unfortunately for the physical applications the number s proves usually 

 to be imaginary or complex, and Jacobi's expression is useless; Legendre calls 

 this the circular form of the E. I. Ill, the logarithmic or hyperbolic form corre- 

 sponding to real s. However, the complete E. I. Ill between the limits o < JTT, 

 or o <u <K, o <e <i, can always be expressed by the E. 1. 1 and II, as Legendre 

 pointed out. 



11.6. The standard forms are given above to which an elliptic integral must be 

 reduced when the result is required in a numerical form taken from the Tables. 

 But in a practical problem the integral arises in a general algebraical form, and 

 theory shows that the result can always be made, by a suitable substitution, to 

 depend on three differential elements, of the I, II, III kind, 



* 



Vs 



TT / \ ds 



II (s-a}- .. 



where S is a cubic in the variable 5 which may be written, when resolved into 

 three factors, 



S = 4'S Si'S - Sz'S Ss 



in the sequence <x>si>s 2 >s 3 > - ex, and normalised to a standard form of 

 zero degree these differential elements are 



II - _ 



\Ai $3 V o 



III 



S- (T 



S denoting the value of S when s = cr. 



The relative positions of 5 and (7 in the intervals of the sequence require 

 preliminary consideration before introducing the Elliptic Functions and their 

 notation. 



