Page:Calculus Made Easy.pdf/176



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



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

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