424 
ME. E. AY. BARNES ON INTEGRAL FUNCTIONS. 
among the parameters that the region near infinity is not ultimately a lacunary 
space. The parameters are, of course, the constants which enter into the expression 
of the general zero in the form 
«nl, „2 • • • = <f> { n U «2> • • •)• 
Sucli functions have been scarcely considered in analysis. The ?t ple gamma function 
is the most simple example which it is possible to give. The theory requires develop¬ 
ment, since from quotients of multiple integral functions can be built up the general 
solution of a linear difference equation. 
It is to be noticed that, by the coalescence of the parameters, multiple integral 
functions give rise to functions of lower sequence with repeated zeros. Thus the 
function^ 
- z 1 -by co 
G( 2 ) = F ^(2^ e -*--- n 
arises from the double gamma function when the parameters w 1 and cj. 2 each become 
equal to unity. 
The separation of multiple functions into functions with repeated and non-repeated 
zeros and their classification would be carried out on parallel lines to the process 
adopted for simple functions. As, however, detailed developments of the asymptotic 
expansions of such functions are not investigated in the present memoir, I do not 
intend to consider such functions further. 
It has been already observed that by the substitution of z m (m integral) for z, we 
derive from any simple integral function a function with m times as many sequences 
of zeros. The substitution of e : for z will transform a simple function into one of 
double sequence. [An example of this is given subsequently (§ 62), where Lambert’s 
function is derived from one of simple sequence.] By transformations of greater 
complexity w r e may evidently construct functions of limitless range. 
§ 21. We are still far from exhausting the category of integral functions. For 
instance, w T e may have ring functions, that is to say, functions whose zeros are 
situated on concentric circles : the number of zeros on the circle depending upon n. 
ao 
We can readily see that such a function is given by the product II 
71 = 1 
where y (n) is a function of n which is equal to an integer for all values of n, and 
where, if y (n) = r. inversely n = x(j (r), and 
L t 
00 . 
For, the product will converge with 
n 
n=k 
1 - 
<M«) 
x(<0' 
* See a paper by the author, ‘ Quart. Journ. Math.,’ vol. 31, pp. 264 et seq. 
