Pera, + ce, = — yap 2 Uae ik (ammeter as Wey e = \((25) 
in which the right member expresses the rate at which heat is converted into energy 
of mean-mean-motion, together with the rate at which energy of relative-mean-motion 
is transformed into energy of mean-mean-motion ; while equation (19) shows whence 
the transformed energy is derived. 
The similarity of the parts taken by the transformation of mean-mean-motion into 
relative-mean-motion, and the conversion of mean-motion into heat, indicates that 
these parts are identical in form; or that the conversion of mean-motion into heat is 
the result of transformation, and is expressible by a transformation function similar 
in form to that for relative-mean-motion, but in which the components of relative 
motion are the components of the heat-motions and the density is the actual density 
at each point. Whence it would appear that the general equations, of which equations 
(19) and (16) are respectively the adaptations to the special condition of uniform 
density, must, by indicating the properties of matter involved, atford mechanical 
explanations of the law of universal dissipation of energy and of the second law of 
thermodynamics. 
The proof of the existence of a criterion as obtained from the equations is quite 
independent of the properties and mechanical principles on which the effect of the 
variations of mean-mean-motion on the distribution of relative mean-motion depends. 
And as the study of these properties and principles requires the inclusion of condi- 
tions which are not included in the equations of mean-motion of incompressible fluid, 
it does not come within the purpose of this paper. It is therefore reserved for 
separate investigation by a more general method. 
The Criterion of Steady Mean-motion. 
23. As already pointed out, it appears from the discriminating equation that the 
possibility of the maintenance of a state of relative-mean-motion depends on p/p, the 
variation of mean-mean-motion and the periods of the relative-mean-motion. 
Thus, if the mean-mean-motion is in direction x only, and varies in direction y 
only, if w’, v’, w’ are periodic in directions «, y, z, a being the largest period in space, 
so that their integrals over a distance a in direction are zero, and if the co-efficients 
of all the periodic factors are a, then putting 
+ du/dy=C*,; 
taking the integrals, over the space a of the 18 squares and products in the last 
term on the left of the discriminating equation (24) to be 
