FLUIDS AND THE DETERMINATION OF THE CRITERION. 143 
d — ad = @ == wl 
(G+ in tee te )E a 
f a (D'se + pw’) | +z, > ce (Dye + uv')| + F [ul (D'ye + uw’) 
a {ie al ra (Pye + v'w’)) + = [e(py + ev + 5 Li (ep oun | 
L she < [w’ ( Pez + ww)| + - [w’ (xa +w'r’)| + <[w (pz + w'w')| J 
dy 
a 
dus 7! A du’ 
13 p TF dx a De dz 

, du , dul , du’ f- —— Oe 0 6) 
Paz g ie + Piyz ap (Dee a | puu In b PUY i we 
| 
r 
| 
| 
J 
i dv’ dv’ 
+ dv’ mek ae , LAT 
| ae 2 Py dz + Pw dy te Pix dz 
R 
where only the mean values, over the space §,, of the expressions in the right member 
are taken into account. 
This is the equation for the mean rate, over the space §,, of change in the energy 
of relative-mean-motion per unit of volume. 
It may be noticed that the rate of change in the energy of mean-mean-motion, 
together with the mean rate of change in the energy of relative-mean-motion, must 
be the total mean-rate of change in the energy of mean-motion, and that by adding 
the equations (17) and (19) the result is the same as is obtained from the equation (3) 
of energy of mean-motion by omitting all terms which have no mean value as summed 
over the space §). 
The Expressions from Transformation of Energy from Mean-mean-motion to Relative- 
mean-motion. 
16. When equations (17) and (19) are added together, the only expressions that 
do not appear in the equation of mean energy of mean-motion are the last terms on 
the right of each of the equations, which are identical in form and opposite in sign. 
These terms which thus represent no change in the total energy of mean-motion 
can only represent a transformation from energy of mean-mean-motion to energy of 
relative-mean-motion. And as they are the only expressions which do not form part 
of the general expression for the rate of change of the mean energy of mean-motion, 
they represent the total exchange of energy between the mean-mean-motion and the 
relative-mean-motion. 
It is also seen that the action, of which these terms express the effect, is purely 
