184 Proceedings of the Koyal Society of Edinburgh. [Sess. 
a particular case, Fourier’s series) just as radiation is analysed by the 
spectroscope. 
Consider first a single one of these periodic constituents, say 
A sin Xx , 
where A and X are constants. We can without loss of generality suppose 
X to be positive. The period of this term is 27 t/X. We shall now show 
that if this period is less than 2w, then an expression can he found which 
is cotahular with the given term and which has a period greater than 2w. 
For, e.g., if the period lies between 2w and 2^^;/3, so that X lies between 
Trjw and SttIw, the function 
A sin 
has the same values as AsinXcc when x = a, a + w, a — w, a + 2w, etc.: and 
since X lies between tt/'u; and Sirlw, we see that (X — 27tIw) lies between 
— irlw and tt/'Iu, so the period of this new term is greater than 2w. 
Similarly if the period of the given term lies between 2w/S and 2w'5, so 
that X lies between Stt/w and birlw, then the function 
is cotabular with A sin Xx and has a period greater than 2w. Other 
possibilities can be treated in the same way, and the theorem stated is thus 
established. 
We are thus led to the idea that if a function is given which can be 
analysed by Fourier’s integral-theorem (or Fourier’s series) into periodic 
constituents, then we can find another function which is cotabular with it 
and which has no constituents of period less than 2w. That is to say, we 
can replace the given function hy a cotabular function in such a way as 
to remove all the rapid oscillations from it. 
§ 4. Introduction of the cardinal function. 
We shall now carry out what has been indicated in the preceding 
article, namely, to analyse a given function into a number (generally an 
infinite number) of periodic constituents, then to replace the short-period 
components by long-period components which are cotabular with them, 
and finally to synthetise all the components into a new function. It will 
be shown later that this new function, which will be called the cardinal 
function, has certain remarkable properties. 
Let/(cc) be the given function, from which all infinities except for 
