4 PROFESSOR ANDREW GRAY, 



If there be endless tubes of flow contained entirely within the surface, no change 

 in the surface integral in this result is required. The volume integral on the right, 

 however, must be evaluated for each such tube. 



The expression 



dq q da- 

 ds o- 3s 



is the "divergence" of the fluid within a tube of flow, that is, the time-rate of 

 increase of the volume of the fluid, per unit of volume, at the element ds. This 

 expression is equivalent to 



du dv dw 



dx dy r)z' 



the divergence expressed in terms of the component velocities with reference to 

 rectangular axes. [It is not difficult to verify this equivalence.] 

 Equation (5) may thus be written in the form 



(T + E) = fp(lu + mv + nw)dS + j f fpP~ + ^ + d ^\dxdydz . (6), 



where I, m, n are the direction-cosines of the inward drawn normal to the element 

 of surface rfS. 



8. From the equation (5) [or (6)] we can deduce the value of 



f(yq 2 + pV)dn, 



that is, of the time-rate of variation of the energy within the fixed surface S. 



Let U denote the volume of the space within S, then this at time t is the volume 



of the mass of fluid which we have been considering. Let also e denote the average 



value of the sum of the kinetic and potential energies per unit of volume ; then for the 



whole mass of the fluid 



T + E = Ue. 

 Thus 



therefore 



-(T + E) = e— + U-; 

 dV ' dt dt' 



4r> +E >-4; 



for U, considered as the volume of a definite mass of fluid, is subject to change. This 

 by § 7 is the same thing as the equation 



-f(^/ + ,V)(| + ?|), to . (7). 



But by >j 2 



|< W + P V) = 1(| W 2 + P Y) + q^pq* + P Y). 



