190 Proceedings of the Poyal Society of Edinburgh. [Sess. 
Now it is well known that 
is zero when k>l\ and 
sin y cos ky 
> y 
dy 
sin y sin ky 
y 
dy 
is always zero. Hence in the above repeated integral the first integration 
gives a zero result so long as \w > ir ; that is to say, there are in the 
expression (5) no constituents of the type cos {\{x — iul)} for which \w>7r, 
and for which therefore the period is less than 2w. 
The theorem being thus seen to be true for every single term 
of the series (3), is consequently true for the cardinal function as 
a whole. 
We may remark in passing that it is possible to construct an infinite 
number of functions cotabular with f(x) by means of series more or less 
resembling the series (3) : for instance, the function 
sin— (x -a- rw) 
n 
w 
r 
1 
1 
where c denotes any real positive constant, and m and n denote any 
positive integers, is a function cotabular with f(x). But this function 
does not possess the property characteristic of the cardinal function, 
namely, that periodic constituents of period less than 2w are absent. 
Such functions are, however, all of them solutions of the problem 
“ To find an analytical expression for a function when we know the 
values which it has for the values a, a-fw, a — w, a + 2w ... of its 
argument ” ; which is essentially the fundamental problem of the theory 
of interpolation. 
§ 7. Solution of the questions proposed m § 1. 
We are now in a position to answer the first of the questions proposed 
in § 1, as to which of the functions of the cotabular set is represented by the 
expansion 
/u -t + 9T— oYo + 31 67 i + 
4! 
The answer is that this expansion represents the cardinal function. This 
we shall now prove. 
