23] 
ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
135 
We derive from the above the somewhat singular conclusion, that the complete 
functions are not absolutely determinate functions of the modulus ; notwithstanding 
that they are given by the apparently determinate conditions, 
l 
dx 
— cV) (1 + e 2 x 2 ) ’ 
l 
dx 
J (1 + c*x 2 ) (1 — e 2 x?) 
In fact definite integrals are in many cases really indeterminate, and acquire 
different values according as we consider the variable to pass through real values, or 
through imaginary ones. Where the limits are real, it is tacitly supposed that the 
variable passes through a succession of real values, and thus a>, v may be considered 
as completely determined by these equations, but only in consequence of this tacit 
supposition. If c and e are imaginary, there is absolutely no system of values to be 
selected for w, v in preference to any other system. The only remaining difficulty is 
to show from the integral itself, independently of the theory of elliptic functions, that 
such integrals contain an indeterminateness of two arbitrary integers; and this difficulty 
is equally great in the simplest cases. Why, d priori, do the functions 
contain a single indeterminate integer ? 
Obs. I am of course aware, that in treating of the properties of such products as 
absolutely necessary to pay attention to the relations between 
the infinite limiting values of m and n; and that this introduces certain exponential 
factors, to which no allusion has been made. But these factors always disappear from 
the quotient of two such products, and to have made mention of them would only 
have been embarrassing the demonstration without necessity.