156 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
(1) The boundary conditions 
= \0) wheney) == Dy a yee ne ee CL 
(2) The equation of continuity 
dtd = OFF) aa a ee CO 
(3) The first of the equations of motion (38) 
d 
PD. ( au du fae ——, | : 
de ONG ae 73) p ley (wv’) + is (ww ) Peart (45)) 3 


or putting 
u=U+u—U and dp/dz =p U/dy? 
as in the singular solution, equation (46) becomes 

1? u—U) @ (& —U) 1 FF jy 
al a Se )=p 15, 0) + 4, ways aR SoS eG (47) ; 
dy? de 
(4) The integral of (47) over the section of which the left member is zero, and 
the mean value of » du/dy = »dU/dy wheny=+b,. . . (48). 
From the condition (3) it follows that if w is to be symmetrical with respect to the 
boundary surfaces, the relative-mean-motion must extend throughout the tube, so 
that 
> del , dwlw\ 4 . : 
5 7 lz Sa ay? . . . . . . 
le ay sles Je is a function of y (49). 
And as this condition is necessary, in order that the equations (38) of mean-mean- 
motion and the equations (16) of relative-mean-motion may be satisfied for steady 
mean-motion, it is assumed as one of the conditions for which the criterion is sought. 
The components of relative-mean-motion must satisfy the periodic conditions as 
expressed in equations (12), which become, putting 2c for the limit in direction z, 
re 
) 
‘ [7 (er | dx = | w’ dx = 0 
~'0 “0 “0 
IP [ u' dy dz = 0 
(2) The equation of continuity 
du'/da + dv'/dy -+ dw'/dz = 0. 
