FLUIDS AND THE DETERMINATION OF THE CRITERION. 145 
Nate eel oe 
ee a — 0 = 
puu — + puv a + pu'w ree. 
—— dw --7 dw —--, dw 
=- zen a 
Luly), i a | 
— ({| 4 aF pow = + pLr z + pv Ww = dx dy dz 
L + pw'w ae + pw Ai + pw'w' — i | 
(i A Cae du! , du’) 
Pee FZ + Bea + Px a, | 
dv’ 
, pe | 
+ {IX tee a 2 WY dy aD erga NS ChaChjdie). Te ek Tg ak (XO), 
, aw! du’ 
LP * de +P, iy ot Ps GEE 3) 

This equation expresses the fundamental relations :— 
(1) That the only integral effect of the mean-mean-motion on the relative-mean- 
motion is the integral of the rate of transformation from energy of mean-mean- 
motion to energy of relative-mean-motion. 
(2) That, unless relative energy is altered by actions across the surface within which 
the integration extends, the integral energy of relative-mean-motion will be increasing 
or diminishing according as the integral rate of transformation from mean-mean- 
motion to relative-mean-motion is greater or less than the rate of conversion of the 
energy of relative-mean-motion into heat. 
19. For p’,,, &c., are substituted their values as determined according to the 
theory of viscosity, the approximate truth of which has been verified, as already 
explained. 
Putting 
eRe (ODS, 
we have, substituting in the last term of equation (20), as the expression for the 
rate of conversion of energy of relative-mean-motion into heat, 
d du’) | dv’ | da’ 
_ {({ = (pH) dz dydz= ({f [2 (s a = =| 
6 du \2 du {dv'\? dw’\? | 
—»{\- oe $+ ) +2((7) +(7) + (a) | 
du’ | d\2 dw’ dw’? dv’ du’ ; 
aE (a a5 7) a GG a iE) v ie a d nl ‘| Lae dyad: se (2) 


MDCCUXCV.—A. U 
