FLUIDS AND THE DETERMINATION OF THE CRITERION. 159 
Then substituting for the values of w’, v’, w’ from equations (53), and integrating in 
direction a over 27//,and omitting terms the integral of which, in direction y, vanishes 
by the boundary conditions, 
[fo (ray ae = 2 ffs{epr (ae + 2) + 2 (ne [(S2) + (FY) 
(GN? 1 (LEN oh aly 5 
+ (Se) + (Ge) haya ines ee ae (7h) 

In a similar manner, substituting for w’ v’, integrating, and omitting terms which 
vanish on integration, the rate of transformation of energy from mean-mean-motion, 
as expressed by the left member in the discriminating equation (43), becomes 
5 ds ; dB, la,\ du 
ez v a dy dz = 4 {(s E (1. = = Be: ) Al DCB: ~ 3 > (G3) 
dy] dy 


And, since by Art. 31, conditions (3) equation (47), 
AO IO) se py, We) - Sa iice te Pom ey eet oe) 
ae ly 
integrating and sae the boundary conditions, 
bay G30) ape poo) =p [arr ly ee era(60) 
And since at the boundary u — U is zero, 
AITO O oS a oe (GN), 
Whence, putting U-++-w—U for w in the right member of equation (58), substituting 
for 1 — U from (60), integrating by parts, and remembering that 

PU Ii ° . rac 
ie ae 3 a WHOA HY 1S COMA 4 oe. oo (OX), 
also that 

uv =t {nt (na — Bee) f ile ee meee (OS): 

‘ IB. Lat, \® 
=3|405 i 5 [dy [nt (cn = B.= a ett es = | (al) Cae Bia dy] . (64). 
by J —by 
