XIV

example, the above graph represents the size at any time of a population with a uniform birth-rate, a uniform death-rate, and no emigration, where the $x$ corresponds to the time reckoned from any convenient day, and the $y$ represents the population to the proper scale. The scale must be such that $OA$ represents the population at the date which is taken as the origin. But we have here come upon the idea of "rates of increase" which is the topic for the next chapter.

An important function nearly allied to the

exponential function is found by putting $-x^2$ for $x$ as the argument in the exponential function.

We thus get $\exp (-x^2)$. The graph $y=\exp (-x^2)$ is given in [fig.]30. 30

The curve, which is something like a cocked hat, is called the curve of normal error. Its

corresponding function is vitally important to the theory of statistics, and tells us in many cases the sort of deviations from the average results which we are to expect.

Another important function is found by combining the exponential function with the sine, in this way:–- $y=\exp (-cx)\times \sin \frac{2\pi x}{p}.$ 31

Its graph is given in [fig.]31. The points $A$, $B$, $O$, $C$, $D$, $E$, $F$, are placed at equal intervals $\frac{1}{2}p$, and an unending series of them should be drawn forwards and backwards. This function represents the dying away of vibrations under the influence of friction or of "damping" forces. Apart from the friction, the vibrations would be periodic, with a period $p$; but the influence of the friction

makes the extent of each vibration smaller than that of the preceding by a constant percentage of that extent. This combination of the idea of "periodicity" (which requires

the sine or cosine for its symbolism) and of "constant percentage" (which requires the exponential function for its symbolism) is the reason for the form of this function, namely, its form as a product of a sine-function into an exponential function.