FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. H 
§ 5. Combining II. with III. it appears that when both the density and the vis 
viva are subject to change that the elasticity is equal to their product, or e = A'V’, 
and this is the law that includes all the conditions of equilibrium of an enclosed 
homogeneous medium. One other condition only remains to be specified. Under a 
constant pressure the density is inversely as the vis viva or mean square molecular 
velocity (A 3 == if e is constant).IV. 
§ 6. In concluding this part of the subject, we cannot fail of being sensible of the 
analogies that subsist between these synthetical deductions and the chief properties 
that distinguish aeriform fluids. 
The first point that was inductively established is Mariotte’s law, viz. : at the 
same temperature the density of air is as its compression. This is analogous to the 
second deduction :—The square of the velocity being constant, the elastic force of a 
medium is proportional to its density. The accordance appears as complete as could 
be desired, and there is a residual evidence in favour of v 2 being identical with 
temperature, or being a quality that varies simultaneously with it. 
The second point is Dalton and Gay-Lussac’s law of expansion. By experimenting 
upon the same weight of air at different temperatures under a constant pressure, these 
philosophers found that an increment of one degree caused always the same 
augmentation of bulk, and that this amounted to -j^-gth part of the space that it 
occupied at 32°. Thus, if the same law hold good at all temperatures, 480 cubic 
inches of air at this temperature should diminish one inch in bulk for every degree 
it was lowered in temperature, and would become zero in bulk at 480° below the 
freezing point of water, or ~ 448° on Fahrenheit scale. 
Now in IV. we had A 3 = —, or v 3 == -r, when e is constant; but — is the volume 
• v 2 ‘A 3 A 3 
occupied by a constant number of molecules; hence with the same constant number 
of molecules the volume is as the mean square molecular velocity, and a constant 
increment of vis viva is followed by the same increment of volume under a constant 
pressure, and as the constant increment of volume (1 cubic inch) is to the constant 
increment of vis viva (1°) so is the volume (480 cubic inches) corresponding to a 
certain vis viva (32° Fahr.) to that vis viva (480°). 
The analogy therefore still holds good, and the evidence continues in favour of the 
absolute temperature being represented by v 3 . 
When air is not allowed to expand and heat is applied, the elastic force increases 
with the temperature, and a rise of 1° causes an absolute increase in the elasticity, 
which is the same at all temperatures, and corresponds with the increase of bulk it 
would assume if allowed freely to expand. This is analogous to III., where, the 
density being constant or bulk unchanged, the elasticity is shown to be proportional 
to the mean square molecular velocity. 
Thus, the laws of Mariotte and of Dalton and Gay-Lussac are represented by 
