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,

$Q_{t}=Q_{0}\epsilon ^{-at}$,

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

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