128 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
and relative-motion may be distinct. These conditions as to the distribution of € &c., 
are, however, obviously satisfied in the case of heat motion, and do not present 
themselves otherwise in this paper. 
From the definite geometrical basis thus obtained, and the definite expressions 
which follow for the condition of distribution of uw, &c., under which the method of 
analysis is strictly applicable, it appears that this method may be rendered generally 
applicable to any system of motion by a slight adaptation of the meaning of the 
symbols, and that it does not necessitate the elimination of the internal motion of 
the molecules, as has been the custom in the theory of gases. 
Taking wu, v, w to represent the motions (continuous or discontinuous) of the matter 
passing a point, and p to represent the density at the point, and putting w, &c., for 
the mean-motion (instead of w as above), and w’, &c., for the relative-motion (instead 
of €as before), the geometrical conditions as to the distribution of u, &c., to satisfy 
the conditions of mean-momentum and mean-energy are, substituting p for M, of 
precisely the same form as before, and as thus expressed, the theorem is applicable to 
any mechanical system however abstract. 
(1) In order to obtain the conditions of distribution of molar-motion, under which 
the condition of mean-momentum will be satisfied so that the energy of molar-motion 
may be separated from that of the heat-motion, wu, &c., and p are taken as referring to 
the actual motion and density at a point in a molecule, and §, is taken of such 
dimensions as may correspond to the scale, or periods in space, of the molecular 
distances, then the conditions of distribution of wu, under which the condition of mean- 
momentum is satisfied, become the conditions as to the distribution of molar-motion, 
under which it is possible to distinguish between the energies of molar-motions and 
heat-motions. 
(2) And, when the conditions in (1) are satisfied to a sufficient degree of approxi- 
mation by taking w to represent the molar-motion (w in (1)), and the dimensions of 
the space 8 to correspond with the period in space or scale of any possible periodic or 
eddying motion. The conditions as to the distribution of u, &c. (the components of 
mean-mean-motion), which satisfy the condition of mean-momentum, show the 
conditions of mean-molar-motion, under which it is possible to separate the energy 
of mean-molar-motion from the energy of relative-molar- (or relative-mean-) motion 
Having thus placed the analytical method used in the kinetic theory on a definite 
geometrical basis, and adapted so as to render it applicable to all systems of motion, 
by applying it to the dynamical theory of viscous fluid, I have been able to show :— 
Feb. 18, 1895.] 
(a) That the adoption of the conclusion arrived at by Sir GABRIEL Stoxes, that the 
dissipation function represents the rate at which heat is produced, adds a definition 
to the meaning of w, v, w—the components of mean or fluid velocity—which was 
previously wanting ; 
