FLUIDS AND THE DETERMINATION OF THE CRITERION. 147 
Thus the second conclusion of Art. 14, and the whole of the previous argument is 
verified, and the properties of matter which prevent the maintenance of mean-motion 
with periods of the same order of magnitude as those of the heat-motion are shown to 
be amongst those properties of matter which are included in the equations of motion 
of which the truth has been verified by experience. 
The Cause of Transformation. 
21. The transformation function, which appears in the equations of mean-energy of 
mean- and relative-mean-motion, does not indicate the cause of transformation, but 
only expresses a kinematical principle as to the effect of the variations of mean-mean- 
motion, and the distribution of relative-mean-motion. In order to determine the 
properties of matter and the mechanical principles on which the effect of the variations 
of the mean-mean-motion on the distribution and angular distribution of relative-mean- 
motion depends, it is necessary to go back to the equations (16) of relative-momentum 
at a point; and even then the cause is only to be found by considering the effects of 
the actions which these equations express in detail. The determination of this cause, 
though it in no way affects the proofs of the existence of the criterion as deduced from 
the equations, may be the means of explaining what has been hitherto obscure in the 
connection between thermodynamics and the principles of mechanics. That such may 
be the case, is suggested by the recognition of the separate equations of mean- and 
relative-mean-motion of matter. 
The Equation of Energy of Relative-mean-motion and the Equation of 
Thermodynamics. 
22. On consideration, it will at once be seen that there is more than an accidental 
correspondence between the equations of energy of mean- and relative-rmean-motion 
respectively and the respective equations of energy of mean-motion and of heat in 
thermodynamics. 
If instead of including only the effects of the heat-motion on the mean-momentum 
as expressed by p,., &e., the effects of relative-mean-motion are also included by 
putting p., for p,. + pw’, &e., and p,, for p,. + pw’, &e., in equations (15) and (17), 
the equations (15) of mean-mean-motion become identical in form with the equations 
(1) of mean-motion, and the equation (17) of energy of mean-mean-motion becomes 
identical in form with the equation (3) of energy of mean-motion. 
These equations, obtained from (15) and (17) being equally true with equations (1) 
and (3), the mean-mean-motion in the former being taken over the space S, instead of 
S, as in the latter, then, instead of equation (9), we should have for the value of the 
last term— 
U 2 
