194 
Proceedings of the Royal Society of Edinburgh. [Sess. 
dense on the segment of the real axis between —1 and +1). In such cases 
the coefficients a^, a 2 , .... of the expansion can be determined in terms 
of f{x), and there is no doubt as to what function is represented by the 
expansion so long as it converges — there is nothing analogous to the 
property of cotabularity. When, however, the roots of the polynomials, 
instead of being everywhere-dense on a segment, are distributed discretely 
over the whole infinite length of the real axis of x (as is the case in 
the expansion 
/o + i + 
- 1) _^(w+l)n(n-l) 
2 ! 
^ 8 % + ^ 
3! 
where the polynomials are 1, n, n(n—l), (n-i-l)n(n — 1), etc.), it seems 
probable that a property analogous to cotabularity will come into evidence, 
and the theory of the expansion will depend essentially on a “ cardinal 
function ” analogous to that introduced above. 
The results of the present paper suggest another development. For 
long past the applied mathematicians have complained that Pure 
Mathematics is daily becoming more complicated and harder to understand. 
This complaint refers chiefiy to the increased rigour with which the 
theories of Analysis are now expounded, and which is closely connected 
with the extension of knowledge regarding discontinuities, singularities, 
and other phenomena of which the older mathematics took no account. 
Indeed, the modern Theory of Functions of a Real Variable is concerned 
largely with cases in which the distribution of fluctuations and singularities 
transcends all intuitive or geometrical representation. It seems possible 
that some of the difficulties of such cases might be avoided by the 
introduction of a function analogous to the “ cardinal function ” of the 
present paper, which would be simpler than the function under discussion, 
but would be equal to it for an infinite number of values of the variable, 
and could be substituted for it in all practical and some theoretical 
investigations. 
{Issued separately July 13, 1915.) 
