144 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
kinematical, depending simply on the instantaneous characters of the mean- and 
relative-mean-motion, whatever may be the properties of the matter involved, or the 
mechanical actions which have taken part in determining these characters. The 
terms, therefore, express the entire result of transformation from energy of mean- 
mean-motion to energy of relative-mean-motion, and of nothing but the transforma- 
tion, Their existence thus completely verifies the first of the general conclusions 
in Art. 14, 
The term last but one in the right member of the equation (17) for energy of 
mean-mean-motion expresses the rate of transformation of energy of heat-motions 
to that of energy of mean-mean-motion, and is entirely independent of the relative- 
mean-motion. 
In the same way, the term last but one on the right of the equation (19) for 
energy of relative-mean-motion expresses the rate of transformation from energy of 
heat-motions to energy of relative-mean-motion, and is quite independent of the 
mean-mean-motion. 
17. In both equations (17) and (19) the first terms on the right express the rates 
at which the respective energy of mean- and relative-mean-motion are increasing 
on account of work done by the stresses on the mean- and _ relative-motion 
respectively, and by the additions of momentum caused by convections of relative- 
mean-motion by relative-mean-motion to the mean- and _ relative-mean-motions 
respectively. 
It may also be noticed that while the first term on the right in the equation (19) 
of energy of relative-mean-motion is independent of mean-mean-motion, the corre- 
sponding term in equation (17) for mean-mean-motion is not independent of relative- 
mean-motion. 
A Discriminating Equation. 
18. In integrating the equations over a space moving with the mean-mean-motion 
of the fluid the first terms on the right may be expressed as surface integrals, which 
integrals respectively express the rates at which work is being done on, and energy 
is being received across, the surface by the mean-mean-motion, and by the relative- 
mean-motion. 
If the space over which the integration extends includes the whole system, or such 
part that the total energy conveyed across the surface by the relative-mean-motion is 
zero, then the rate of change in the total energy of relative-mean-motion within the 
space is the difference of the integral, over the space, of the rate of increase of this 
energy by transformation from energy of mean-mean-motion, less the integral rate 
at which energy of relative-mean-motion is being converted into heat, or integrating 
equation (19), 
