426 MR W. J. M. RANKINE ON THE CENTRIFUGAL THEORY OF ELASTICITY, 
tion, that the atomic atmospheres might be treated in calculation as if spherical, 
did not give rise to any error. 
By the aid of certain transformations in those equations, I have been enabled, 
in investigating the principles of the mutual transformation of heat and expansive 
power, to deduce JouLn’s daw of the equivalence of heat and mechanical power 
directly from them, instead of taking it (as I did in my previous papers) as a con- 
sequence of the principle of vis-viva. Carnot’s law is also deduced directly from 
the hypothesis, as in one of the previous papers. 
(2.) Classification of Elastic Pressures.—The pressures considered in the present 
paper are those only which depend on the volume occupied by a given weight of 
the substance ; not those which resist change of figure in solids and viscous liquids. 
Certain mathematical relations exist between those two classes of pressures; but 
they do not affect the present investigation. 
To illustrate this symbolically, let V represent the volume occupied by unity 
of weight of the substance, so that ~ is the mean density; Q, the quantity of heat 
in unity of weight, that is to say, the vis-viva of the molecular revolutions, which, 
according to the hypothesis, give rise to the expansive pressure depending on heat ; 
and let P denote the total expansive pressure. Then, 
PEF (VQ) 4 OO) widentos 00) aunsisd Hades 
In this equation, F (V, Q) is the pressure of the atomic atmospheres at the sur- 
faces called their boundaries, which varies with the centrifugal force of the mole- 
cular vortices as well as with the mean density; and /(V) is a portion of pressure 
due to the mutual attractions and repulsions of distinct atoms, and varying with 
the number of atoms ina given volume only. If the above equation be differentiated 
with respect to the hyberbolic logarithm of the density, we obtain the coefficient 
of elasticity of volume 

1 dP d d E 
Dirt sigh Vaan —av FW. @)-ayF™ ° 2 ° (1 A.) 
“via We Vv 
where 3 denotes the cubic compressibility. 
The latter portion of this coefficient, 4 J (V), consists of two parts, one of 
Ae 
which is capable of being resolved into forces, acting along the lines joining the 
atomic centres, and gives rise to rigidity, or elasticity of figure, as well to elas- 
ticity of volume, while the other, which is not capable of being so resolved, gives 
rise to elasticity of volume only. The ratio of each of those parts to their sum 
must be a function of the heat, the former part being greater, and the latter less, 
as the atomic atmosphere is more concentrated round the nucleus; that is to say, 

