12 THE EQUATIONS OF MOTION. [CHAP. I 



fore call E the intrinsic energy of the fluid, per unit mass. Now 

 recalling the interpretation of the expression 



du dv dw 

 dx dy dz 



given in Art. 7 we see that the volume-integral in (4) measures the 

 rate at which the various elements of the fluid are losing intrinsic 

 energy by expansion* ; it is therefore equal to DWjDt, 



where W = fffEpda;dydz, (8). 



Hence -^ (T + V+ W) = ! fp (lu + mv + nw) dS (9). 



The total energy, which is now partly kinetic, partly potential in 

 relation to a constant field of force, and partly intrinsic, is therefore 

 increasing at a rate equal to that at which work is being done on 

 the boundary by pressure from without. 



Impulsive Generation of Motion. 



12. If at any instant impulsive forces act bodily on the 

 fluid, or if the boundary conditions suddenly change, a sudden 

 alteration in the motion may take place. The latter case may 

 arise, for instance, when a solid immersed in the fluid is suddenly 

 set in motion. 



Let p be the density, u, v, w the component velocities immedi 

 ately before, u , v , w those immediately after the impulse, X , Y f , Z 

 the components of the extraneous impulsive forces per unit mass, 

 OT the impulsive pressure, at the point (x, y, z). The change of 

 momentum parallel to x of the element defined in Art. 6 is then 

 pxyz(u ii)\ the ^--component of the extraneous impulsive 

 forces is p&xy&zX , and the resultant impulsive pressure in the 

 same direction is dv/dx . Sx&y&t. Since an impulse is to be 

 regarded as an infinitely great force acting for an infinitely short 

 time (r, say), the effects of all finite forces during this interval are 

 neglected. 



* Otherwise, 



(du dv dw 

 T x + Ty + -S 



