787] 
FUNCTION. 
531 
m and n having any positive or negative integer values whatever, including zero, except 
that m, n must not be simultaneously = 0, these values being taken account of in the 
factor u outside the product. But until further defined, such a product has no definite 
value, and consequently no meaning whatever. Imagine m, n to be coordinates, and 
suppose that we have, surrounding the origin, a closed curve having the origin for its 
centre, i.e. the curve is such that, if a, ¡3 be the coordinates of a point thereof, then 
— a, — /3 are also the coordinates of a point thereof; suppose further that the form 
of the curve is given, but that its magnitude depends upon a parameter h, and that 
the curve is such that, when h is indefinitely large, each point of the curve is at an 
indefinitely large distance from the origin; for instance, the curve might be a circle 
or ellipse, or a parallelogram, the origin being in each case the centre. Then if in 
the double product, taking the value of h as given, we first give to m, n all the 
positive or negative integer values (the simultaneous values 0, 0 excluded) which corre 
spond to points within the curve, and then make h indefinitely large, the product will 
thus have a definite value; but this value will still be dependent on the form of the curve. 
Moreover, varying in any manner the form of the curve, the ratio of the two values 
of the double product will be = exp /3u 2 , where /3 is a determinate value depending only 
on the forms of the two curves; or, what is the same thing, if we first give to the 
curve a certain form, say we take it to be a circle, and then give it any other form, 
the product in the latter case is equal to its former value multiplied by exp ¡3u 2 , 
where /3 depends only upon the form of the curve in the latter case. 
Considering the form of the bounding curve as given, and writing the double 
product in the form 
u + meo + nv 
the simultaneous values m = 0, n = 0 being now admitted in the numerator, although 
still excluded from the denominator, then if we write for instance u-f 2o> instead of u, 
each factor in the numerator is changed into a contiguous factor, and the numerator 
remains unaltered, except that we introduce certain factors which lie outside the 
bounding curve, and omit certain factors which lie inside the bounding curve; we, in 
fact, affect the result by a singly infinite series of factors belonging to points adjacent 
to the bounding curve ; and it appears on investigation that we thus introduce a con 
stant factor exp 7 (m + «). The final result thus is that the product 
does not remain unaltered when u is changed into u + 2&>, but that it becomes there 
fore affected with a constant factor, exp y (w + eo). And similarly the function does not 
remain unaltered when u is changed into u + 2v, but it becomes affected with a factor, 
exp 8 (u + v). The bounding curve may however be taken such that the function is 
unaltered when u is changed into u + 2a> : this will be the case if the curve is a 
rectangle such that the length in the direction of the axis of m is infinitely great in 
comparison of that in the direction of the axis of n ; or it may be taken such that 
the function is unaltered when u is changed into u + 2v: this will be so if the curve 
67—2