XIII

drawing the axis of $OY$ through the point on $OX$ marked ${\displaystyle \frac{\pi }{2}}$, instead of drawing it in its actual position on the figure.

It is easy to construct a `sine' function in 28 which the period has any assigned value $a$. For we have only to write $y=\sin \frac{2\pi x}{a},$ and then $\sin \frac{2\pi (x + a)}{a}

= \sin \left(\frac{2\pi x}{a} + 2\pi\right) = \sin \frac{2\pi x}{a}.$ Thus the period of this new function is now $a$. Let us now give a general definition of what

we mean by a periodic function. The function $f(x)$ is periodic, with the period $a$, if (i) for any value of $x$ we have $f(x)=f(x+a)$, and (ii) there is no number $b$ smaller than $a$ such that for any value of $x$, $f(x)=f(x+b)$.

The second clause is put into the definition because when we have $\sin {\displaystyle \frac{2\pi x}{a}}$, it is not only periodic in the period $a$, but also in the periods $2a$ and $3a$, and so on; this arises since $\sin \frac{2\pi (x+3a)}{a}=\sin (\frac{2\pi x}{a}+6\pi )=\sin \frac{2\pi x}{a}.$ So it is the smallest period which we want to get hold of and call the period of the function. The greater part of the abstract theory of periodic functions and the whole of the applications of the theory to Physical Science are dominated by an important theorem called Fourier's Theorem; namely that, if $f(x)$ be a

periodic function with the period $a$ and if $f(x)$ also satisfies certain conditions, which practically are always presupposed in functions suggested by natural phenomena, then $f(x)$ can be written as the sum of a set of terms in the form191

c 0 + c 1 sin ( 2 π x a + e 1 ) + c 2 sin ( 4