158 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 


and 
/ i *) da, d n ‘d, n 18, = 
—_> le + a cos (nla) ++ G + a sin (nl) ) 
v =X {nla, sin (nlx) — nlB,, cos (nlx) } ( ; (53), 
w =X {uly, sin (nlx) — nl 8, cos (nlx) } J) 
w,v, w' satisfy the equation of continuity. And, if 
a=B=y=6= da/dy = dB/dy = dy/dz = dd/dz = 0 when y= + tn (54) 
and «8, ay, «6 are all functions of y? only , 
it would seem that the expressions are the most general possible for the components 
of relative-mean-motion. 
Cylindrical-relative-motion. 
34. If the relative-mean-motion, like the mean-mean-motion, is restricted to 
motion parallel to the plane of ay, 
= 6 = w = 0, everywhere, 
y , Ay 
and the equations (53) express the most general forms for wu’, v' in case of such 
cylindrical disturbance. 
Such a restriction is perfectly arbitrary, and having regard to the kinematical 
restrictions, over and above those contained in the discriminating equation, would 
entirely change the character of the problem. But as no account of these extra 
kinematical restrictions is taken in determining the limit to the criterion, and as it 
appears from trial that the value found for this limit is essentially the same, whether 
the relative-mean-motion is general or cylindrical, I only give here the considerably 
simpler analyses for the cylindrical motion. 
The Functions of Transformation of Energy and Conversion to Heat for Cylindrical 
Motion. 
d ( 1’) 
dt \ / 
35. Putting 

for the rate at which energy of relative-mean-motion is converted to heat per unit of 
volume, expressed in the right-hand member of the discriminating equation (43), 
(\{ (,H’) dx dy dz 
=n (fea) +(G) t+ (G+) +25 eee. 6) 
