182 Proceedings of the Koyal Society of Edinburgh. [Sess. 
In the theory of interpolation certain expansions are introduced in 
order to represent the function f{x), for general values of x, in terms of 
the quantities occurring in the above difference-table. We shall consider 
in particular the expansion 
/„ + nhf^ + ’K”- Dsy^ + (»+!>(” -1) 334 + (n + lMn-l)(«-2) gy^ 
( 2 ) 
which is supposed (when it converges) to represent /((X -P where n can 
have any value. It is obvious, however, that there is no reason a 'priori 
why this expansion should represent f{x) in preference to any other 
function of the set cotabular with f{x)\ and thus two questions arise, 
namely : — 
(1) Which one of the functions of the cotabular set is represented by 
the expansion (2)? 
(2) Given any one function f(x) belonging to the cotabular set, is it 
possible to construct from f(x), by analytical processes, that function of 
the cotabular set which is represented by the expansion (2) t 
These questions are answered in the present paper. It is, in fact, shown 
that there is a certain function belonging to the cotabular set which is 
represented by the expansion (2). This function is named the cardinal 
function of the set, and its properties are investigated. A formula is 
given by which the cardinal function may be constructed when any one 
function of the cotabular set is known. 
§ 2. Removal of singularities from a function, by substituting a 
cotabidar function for it 
We shall first show that x/’f(x) has a singularity at a point c, we can 
find a function cotabular with f(x) which has no singularity at c. 
For suppose first that the singularity is a simple pole, so that f{x) 
becomes infinite in the same way as 
r 
X - c 
near the point c. Then the function 
r sin 
7t{x - a) 
/(*)- 
{x — c) sin ^ 
w 
is cotabular with fix), since the factor sin ^ vanishes at all the 
