132 PROFESSOR 0. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
the direction indicated by the first suffix, in the direction indicated by the second 
suffix. 
Then multiplying these equations respectively by u, v, w, imtegrating by parts, 
adding and putting 
2E for p(w+v? + w’) 
and transposing, the rate of increase of kinetic energy per unit of volume is given by 
[ ~ (pre) +5 5 (ups) +— ~ (pez) ; 
(Gj tuntegt: ar + na 3m > 
+ (wpe) + £ (wpe) + Z(mp.) 
[ Poe ae ht a 
2" + Page FPA PP, ey 
| tea gg tie Gy + Pep | 
The left member of this equation expresses the rate of increase in the kinetic 
energy of the fluid per unit of volume at a point moving with the fluid. 
The first term on the right expresses the rate at which work is being done by the 
surrounding fluid per unit of volume at a point. 
The second term on the right therefore, by the law of conservation of energy, 
expresses the difference between the rate of increase of kinetic energy and the rate 
at which work is being done by the stresses. This difference has, so far as I am 
aware, in the absence of other forces, or any changes of potential energy, been equated 
to the rate at which heat is being converted into energy of motion, Sir GABRIEL 
Stokes having first indicated this* as resulting from the law of conservation of 
energy then just established by JouLs. 
7. This conclusion, that the second term on the right of (3) expresses the rate at 
which heat is being converted, as it is usually accepted, may be correct enough, but 
there is a consequence of adopting this conclusion which enters largely into the 
method of reasoning in this paper, but which, so far as I know, has not previously 
received any definite notice. 
* “Cambridge Phil. Trans.,’ vol. 9, p. 57. 
