FLUIDS AND THE DETERMINATION OF THE CRITERION. 135 
same point the difference is the value of wv’ at this point, whence summing these 
differences over the space S about G at «, y, z, since by definition when summed over 
the space S about G 
5 oe — tills Ooi Sale? Sela © oo oo) 
du 
S (pu') = — {45 [p(@— ey ](Ts) + 32le-9 (Ga) 


du 
| 
+$2lo(e—2)I(Fs) +4eb 1 (aay 
| 
ihatis . 
2 (pw) « a? (de ) b? (du 2 (du 
= (p) is < . da Sli 2 al 4F 9 ( + ke. } 
In the same way if ¥(___) be taken over the interval of time 7 including ¢; and 
for the instant ¢ 




“= oe and pu = pu + pw’; 

then since for any other instant ¢’ 

where 3 [p(t — ¢’)] = 0, and S[p (u —u)] = 0. 
It appears that 
du 
= (pw’) =— =[S p(t — te WP + &e. | 
> (pu! (22 
i a al | a 
Sia) <-l1r a) &e. | 
(8B). 

From equations (84) and (8B), and similar equations for & (pv’) and = (pw’), it appears 
that if 
2 (pu) = = (pu') = = (pw) = 0, 
where the summation extends both over the space S and the interval 7, all the terms 
on the right of equations (84) and (8B) must be respectively and continuously zero, or, 
what is the same thing, all the differential coefficients of u, v, w with respect to 
x, y, z and t of the first order must be respectively constant. 
This condition will be satisfied if the mean-motion is steady, or uniformly varying 
