110 LIMITS OF FUNCTIONS OF AN INTEGRAL VARIABLE [iV 
In other cases <p{n) may be defined by a formula, such as ( — !)”, which 
ceases to define for some values of x at any rate (as here in the case of 
fractional values of x with even denominators, or irrational values). But it 
may be possible to transform the formula in such a way that it does define 
for all values of x. In this case, for example, 
( — l) re = cos mr, 
if n is an integer, and the problem of interpolation is solved by the function 
cos XTT. 
In other cases 0 (x) may be defined for some values of x other than 
positive integers, but not for all. Thus from y—n n we are led to y—x x . 
This expression has a meaning for some only of the remaining values of x. 
If for simplicity we confine ourselves to positive values of x, x x has a 
meaning for all rational values of x, since 
{plq)m=*/{plq)P, 
according to the definition of fractional indices adopted in elementary 
algebra. But when x is irrational x x has (so far as we are in a position to 
say at the present moment) no meaning at all. Thus in this case the 
problem of interpolation at once leads us to consider the question of 
extending our definitions in such a way that x x shall have a meaning even 
when x is irrational. We shall see later on how the desired extension may 
be effected. 
Again consider the case in which 
y = \ .2 ...n=n\ 
In this case there is no obvious formula in x which reduces to n! for x=n, 
as x! means nothing for values of x other than the positive integers. This 
is a case in which attempts to solve the problem of interpolation have led to 
important advances in mathematics. For mathematicians have succeeded in 
discovering a function (the Gamma-function) which possesses the desired 
property and many other interesting and important properties besides. 
45. Finite and infinite classes. Before we proceed further 
it is necessary to make a few remarks about certain ideas of an 
abstract and logical nature which are of constant occurrence in 
Pure Mathematics. 
In the first place, the reader is probably familiar with the 
notion of a class. It is unnecessary to discuss here any logical 
difficulties which may be involved in the notion of a ‘class’: 
roughly speaking we may say that a class is the aggregate or 
collection of all the entities or objects which possess a certain 
property, simple or complex. Thus we have the class of British 
subjects, or red-headed Germans, or positive integers, or real 
numbers.