146 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
in which pw is a function of temperature only ; or since p is here considered as constant, 
Lika (om) = — efeL (ie) + (i+ Cl] +(e ¥a) 
if aw \2 » — du'\2 
ae CG a al a G ate =| hae dy de... . . Can 
da dy 



whence substituting. for the last term in equation (20) we have, if the energy of 
relative-mean-motion is maintained, neither increasing or diminishing, 
Shp | 
7 , du UT 7 7 U 
UC = UC == “we == 
daz: uy dy ats d 
—ST aU 



| —— as — | 
— p{|| +o gE 4 wee | dedyde 


—-,dw , {dw , — 
(LULU ea 
| 
{ + wu eal v a + ww FE 
lu’\?  /dv’\? /dw'\? | 
a] du! de! 
(c) 1 a a5 ( | | 
E w’ dv’ i (2 u! du’ 

ree) 

dz dx 
dv’ du’\2 
+ G aly D) J 
which is a discriminating equation as to the conditions under which relative-mean- 
motion can be sustained. 

ag obra) ) + dudy dz = 0. . (24) 

all | 
| 
20. Since this equation is homogeneous in respect to the component velocities of 
the relative-mean-motion, it at once appears that it is independent of the energy of 
relative-mean-motion divided by the p. So that if p/p is constant, the condition it 
expresses depends only on the relation between variations of the mean-mean-motion 
and the directional, or angular, distribution of the relative-mean-motion, and on the 
squares and products of the space periods of the relative-mean-motion. 
And since the second term expressing the rate of conversion of heat into energy of 
relative-mean-motion is always negative, it is seen at once that, whatscever may be 
the distribution and angular distribution of the relative-mean-motion and the varia- 
tions of the mean-mean-motion, this equation must give an inferior limit for the rates 
of variation of the components of mean-mean-motion, in terms of the limits to the 
periods of relative-mean-motion, and p/p, within which the maintenance of relative- 
mean-motion is impossible. And that, so long as the limits to the periods of relative- 
mean-motion are not infinite, this inferior limit to the rates of variation of the mean- 
mean-motion will be greater than zero. 
