154 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
and as the mean values of functions of w’, v’, w’ are constant in the direction of flow, 
d (ww) d (wu) d (w'u’) 
Gi * da 2 da Ose ee, 
By equations (21) and (37) the equations (15) of mean-motion become 




du ere dp Pu Pu Opa! pO pee, 
i= dk ote +i) —0[ Gorm + germ} 
dv dp ds d —— | ‘ 
Poa eh 1 OO) ae wo) (38). 
dw dp ad —-F7 = 
oS = {eas foro} 
The equation of energy of mean-mean-motion (17) becomes 
a) _ 5b, [a (-di) , @ (oui 4 gas team! | 
Eo ba +H 4 (3 Es Us bapla (uuvj+ > (u ww) 
| | 

dy dy 
du \? du \2 ah, Sein 
fly) +(e) freee ay tee a | J 
Similarly the equation of mean-energy of relative-mean-motion (19) becomes 

dk . y’ , Sap / , apap 
LU (Pe FUE) HO (Ply FOR) HW (Diy + WY] 
t / ‘ sae At / / Serhepey , , seer Tate, 
_ ale (piuctuw)+0 (p', vw) +u' (pe tww’)|] 
#2 ee 38 a) au iG ui (a ap | 1" dz ow da 1 da uy dz 
vag Oh 
— p fare tw a EER Ci ek ehali bo. 5 (Uh 


Integrating in directions y and z between the boundaries and taking note of the 
boundary conditions by which wu, w’, v’, w’ vanish at the boundaries together with the 
integrals, in direction z, of 

d /—~du i Mdyhw. — 
s (w a ee (w'w’)], ae [u’ (p.ctu'w’)], &e., 
the integral equation of energy of mean-mean-motion becomes 
fe inde = — {il “ + p \( y er —p {we v = uw’ = a oh jay dz. (41). 
