[715 
716] 
41 
716. 
AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. 
[From the Messenger of Mathematics, vol. vn. (1878), pp. 27—32.] 
If X be a given quartic function of x, and if u, or for convenience a constant 
multiple au, be the value of the integral 
taken from a given inferior limit to 
the superior limit x; then, conversely, x is expressible as a function of u, viz. it is 
expressible in terms of ^-functions of u, where *su, or say ^r{u, §) (g a parameter 
upon which the function depends), is given by definition as the sum of a series of 
exponentials of u; and it is possible from the assumed equation au = f -, and the 
J v(X) 
definition of Shi, to obtain by general theory the actual formulae for the determination 
of x as such a function of u. 
I propose here to obtain these formulae, in the case where X is a product of 
real factors, in a less scientific manner, by connecting the function ®\u (as given by 
such definition) with Jacobi’s function ®, and by reducing the integral 
by a 
linear substitution to the form of an elliptic integral; the object being merely to 
obtain for the case in question the actual formulae for the expression of x in terms 
of ^-functions of u. 
The definition of or, when the parameter is expressed, (u, $) is 
^u=l (~) s e -^ +2im , 
where s has all positive or negative integer values, zero included, from — oo to + oo 
(that is, from — S to + S, $ = oo); the parameter or (if imaginary) its real part, 
must be positive. 
C. XI. 
6