Page:Reflections on the Motive Power of Heat.djvu/209

corresponding to the temperature t, is known. For, by § 30, equation (6), we have

$v{\tfrac {dq}{dv}}={\tfrac {Ep_{_{0}}v_{_{0}}}{\mu }},$

where dq is the quantity of heat absorbed, when the volume is allowed to increase from v to v + dv; or the quantity evolved by the reverse operation. Hence we deduce

$dq={\tfrac {Ep_{_{0}}v_{_{0}}}{\mu }}{\tfrac {dv}{v}}.$

(8)

Now, ${\tfrac {Ep_{_{0}}v_{_{0}}}{\mu }}$ is constant, since the temperature remains unchanged; and therefore we may at once integrate the second number. By taking it between the limits V' and V, we thus find^[1]

$Q={\tfrac {Ep_{_{0}}v_{_{0}}}{\mu }}\log {\tfrac {V}{V'}},$

(9)

where Q denotes the required amount of heat evolved by the compression from V to P'. This expression may be modified by employing the equations PV = P'V' = p₀v₀ (1 + Et); and we thus obtain

$Q={\tfrac {EPV}{\mu (1+Et)}}\log {\tfrac {V}{V'}}={\tfrac {EP'V'}{\mu (1+Et)}}\log {\tfrac {V}{V'}}.$

(10)

1. The Napierian logarithm of ${\tfrac {V}{V'}}$ is here understood.