142 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
Again, multiplying the equations of mean-mean-motion by u,v, Ww respectively, 
adding and putting 2E = p(w? + v? + w’), we obtain 
p ge lt Peet WHY) + 5, [HD WHY] + EL (Pa | 
<3 + E18 (Dy + BW) + ELE (Py FPO) +2 Dy + BW) JE 
Bare dz 
dae [6 Pat WH) + [Pe +WV)] +H [0 (p+ WW) 

fF dy = da Sida a) ( —-; du a = 
| Px Gi Pyx dy Pex dz 


| = =] | | =e an 
= - dy — dv — - dv : eH) ae — dv 
+ OW ; ee Pape + Ain ee FD nasach TT iL 
4 + Pry ae + Pw a + Py d. | 1 1 Oe m vee dy Oe r eo) 
dy Yy lx 
= dp = Oi. S 20 ed LD) 
Pee: ar Ds ae J Te te OE ee hea dz | 
which is the approximate equation of energy of mean-mean-motion in the same sense 
as the equation (3) of energy of mean-motion is approximate. | 
In a similar manner multiplying the equations (16) for the momentum of relative- 
mean-motion respectively by w’, v’, w’, and adding, the result would be the equation 
for energy of relative-mean-motion at a point, but this would include terms of 
which the mean values taken over the space S, are zero, and, since all corresponding 
terms in the energy of heat are excluded, by summation over the space S, in the 
expression for the rate at which mean-motion is transformed into heat, there is no 
reason to include them for the space §,; so that, omitting all such terms and putting 
On’ = p(u? 4.02 +02)». b . . yey len 
we obtain . 
