Page:Calculus Made Easy.pdf/177

 where $$\theta_0$$ is the original excess of temperature of a hot body over that of its surroundings, $$\theta_t$$ the excess of temperature at the end of time $$t$$, and $$a$$ is a constant—namely, the constant of decrement, depending on the amount of surface exposed by the body, and on its coefficients of conductivity and emissivity, etc.

A similar formula,

is used to express the charge of an electrified body, originally having a charge $$Q_0$$, which is leaking away with a constant of decrement $$a$$; which constant depends in this case on the capacity of the body and on the resistance of the leakage-path.

Oscillations given to a flexible spring die out after a time; and the dying-out of the amplitude of the motion may be expressed in a similar way.

In fact $$\epsilon^{-at}$$ serves as a die-away factor for all those phenomena in which the rate of decrease is proportional to the magnitude of that which is decreasing; or where, in our usual symbols, $$\dfrac{dy}{dt}$$ is proportional at every moment to the value that y has at that moment. For we have only to inspect the curve, Fig. 42 above, to see that, at every part of it, the slope $$\dfrac{dy}{dx}$$ is proportional to the height $$y$$; the curve becoming flatter as $$y$$ grows smaller. In symbols, thus