FLUIDS AND THE DETERMINATION OF THE CRITERION. 141 
point x, y, z. Then since uv, v, w are continuous functions of x, y, z, therefore 
u, v, w, and wu’, v, w’, are continuous functions of x, y, z. And as p is assumed 
constant, the equations of continuity for the two systems of motion are : 
du dv oa du’ . dv’ . dwt 
da di a da aa di dy YG 

=O ¢ s- ae (S)s 
also both systems of motions must satisfy the boundary conditions, whatever they 
may be. 
Further putting Dees &c., for the mean values of the stresses taken over the space 
S, and 
Die = Px ae Pox . ° . . ° ° ° ° . . (14) 
and defining S, to be such that the space variations of uv, v, w are approximately 
constant over this space, we have, putting ww’, &c., for the mean values of the squares 
and products of the components of relative-mean-motion, for the equations of mean- 
mean-motion, 
du d = 1 as — d LSE wes aot: 5 
Pa = 15 (Pat puu + pu wu) + ay (Pus + puw + puv) 
+ 5, (Da + pu + pww)| |. . (19), 
See ee &e. | 
&e. = &e. J 
dz 
which equations are approximately true at every point in the same sense as that in 
which the equations (1) of mean-motion are true. 
Subtracting these equations of mean-mean-motion from the equations of mean- 
motion, we have 
Wes 2 ie par es 
fF {Doo + p (uw + wu) + p(u'u’ —wu')} 
ar ae a = {Pye + p (wo + wr) + p (w'v’ — w’v')} &e., &e. (16), 
1 ae — , ar pais f 
LF, Bia + p (tin! + ui) + p (ul! — 0) | 
which are the equations of momentum of relative-mean-motion at each point. 
