532 
FUNCTION. 
[787 
be a rectangle such that the length in the direction of the axis of n is indefinitely 
great in comparison with that in the direction of the axis of to; but the two con 
ditions cannot be satisfied simultaneously. 
11. We have three other like functions, viz. writing for shortness in, n to denote 
m + \, n + ^ respectively, and (to, n) to denote mm + nv, then the four functions are 
wIM (1 + 
i 1 + fsra) 
nn 1+^ 
(to, n) 
nn 1 + 
(to, n) 
nn 1 + 
(to, n) 
the bounding curve being in each case the same; and, dividing the first three of these 
each by the last, we have (except as to constant factors) the three functions sn u, cn u, dn u; 
writing in each of the four functions u + 2co or u + 2v in place of u, the functions 
acquire each of them the same exponential factor exp y (u + co), or exp 8 (u + v), and 
the quotient of any two of them, and therefore each of the functions sn u, cn u, dn u, 
remains unaltered. 
It is easily seen that, disregarding constant factors, the four d-functions are in 
fact one and the same function with different arguments, or they may be written 
6u, 6 (u + ^w), 6(u + ^v), 6 (u + ^co + %v) ; by what precedes, the functions may be so 
determined that they shall remain unaltered when u is changed into u + 2&>, that is, 
be singly periodic, but that the change u into u+2v shall affect them each with the 
same exponential factor exp 8 (a + v). 
12. Taking the last-mentioned property as a definition of the function 6, it 
appears that 6u may be expressed as a sum of exponentials 
6u = AX exp — (vm 2 4- um), 
CO 
where the summation extends to all positive and negative integer values of to, 
including zero. In fact, if we first write herein u + 2<o instead of u, then in each 
term the index of the exponential is altered by — 2coto, = 2m r in, and the term itself 
thus remains unaltered; that is, 6 (u + 2to) = On. But writing u + 2v in place of u, each 
term is changed into the succeeding term, multiplied by the factor exp — (u + v); in fact, 
making the change in question u into u + 2v, and writing also to — 1 in place of to, 
vm 2 + um becomes v (to — l) 2 + (u + 2v) (to — 1), = vm 2 + um — u — v, and we thus have 
6 (u + 2v) = exp |— (u -f- u)j . 6u. In order to the convergency of the series it is 
irivm 2 
necessary that exp 
should vanish for indefinitely large values of to, and this will 
be so if — be a complex quantity of the form a + /3i, a negative; for instance, this will 
be the case if co be real and positive and v be = i multiplied by a real and positive 
quantity. The original definition of 6 as a double product seems to put more clearly 
in evidence the real nature of this function, but the new definition has the advantage 
that it admits of extension to the d-functions of two or more variables.