Page:Calculus Made Easy.pdf/176

The Die-away Curve.

If we were to take $p$ as a proper fraction (less than unity), the curve would obviously tend to sink downwards, as in Fig. 42, where each successive ordinate is ${\tfrac {3}{4}}$ of the height of the preceding one.

The equation is still

$y=bp^{x}$;

Fig. 42.

but since $p$ is less than one, $\log _{\epsilon }p$ will be a negative quantity, and may be written $-a$; so that $p=\epsilon -a$, and now our equation for the curve takes the form

$y=b\epsilon ^{-ax}$.

The importance of this expression is that, in the case where the independent variable is time, the equation represents the course of a great many physical processes in which something is gradually dying away. Thus, the cooling of a hot body is represented (in Newton’s celebrated “law of cooling”) by the equation

$\theta _{t}=\theta _{0}\epsilon ^{-at}$;