Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/434

heat added or taken away, and mention has been made of a form of calorimeter proposed by the author depending upon this principle.

It is evident that if the principle can be proved as a general proposition as relating to the total heat it is also proved in relation to heat differences, that is heat added or subtracted.

The following proof goes beyond the problem as presented by the calorimeter, and applies generally for an enclosure in which the various portions of the gas are artificially constrained to occupy given positions by any means whatever, including, for example, the case of a wave train or other dynamic disturbance.

Let the enclosure be supposed divided into a number of small equal elements, and, examining firstly the conditions that apply to each small element to which it may be supposed that a quantity of heat $h$ is supplied and distributed uniformly, giving rise to a uniform pressure $P$ and temperature $T,$ we have:—

${\frac {P}{\rho }}=T\times {\mbox{const.}}$

but for a perfect gas

$T={\frac {m}{h}}\times {\mbox{const.}}$

where $m$ is the mass of the contents, hence

${\frac {P}{\rho }}={\frac {h}{m}}\times {\mbox{const.}}$

and since

$\rho ={\frac {m}{l^{3}}}$

$P={\frac {h}{l^{3}}}$

for the element with which we are concerned.

Now,	let	$n=$ the number of elements into which the enclosure is divided.
	„	$P_{1}\ P_{2}\ P_{3}$ etc., be the pressures developed in the different elements to which quantities of heat $h_{1}\ h_{2}\ h_{3}$ have been supplied.
	„	$H=$ the total heat.
	„	$P_{m}=$ the resulting mean pressure.

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