NOTES ON HYDRODYNAMICS, CHIEFLY ON VORTEX-MOTION. 5 



Thus we obtain 



Remembering that {dq/ds + (ql<r)d<r/ds}dvJ = d(q<r)/ds . ds, and integrating the second line 

 on the right by parts as in § 7, we obtain 



j?(hpT + pV)dK = J \$tf + pV +p)q n dS + fp(% + i*£)dt3 . . (8). 



The integral on the left of (8) is the rate at which the sum of the kinetic and 

 potential energies within the surface is increasing ; and the integrals on the right show 

 that this rate is equal to the rate of flow of energy into the space across the surface, 

 together with the rate at which work is done by pressure in consequence of expansion 

 of the fluid within the space and of passage of fluid across the surface. 



9. The value of the divergence found above for a tube of flow leads to a corre- 

 sponding mode of expressing the equation of continuity of the fluid. If p be the 

 density of the fluid at the element of volume dm, then, since the increase of volume in 

 time dt is (dq/ds + q/a- .d<r/ds)dt per unit volume, the mass for this at density p is 

 p(dq/ds + q/cr . da-jds)dt. But since the mass of the element is not changed, for it 

 preserves its identity, this expression must have the value — dp/dt . dt. Thus we 

 obtain the equation of continuity in the form 



dt H \ds a- ds) 



or 



dp dp 



(9). 



Op dp /da q d<r\ 



If we put for the divergence, this equation can be written 



f + P© = (10). 



at 



10. As an example of these results we apply them to the steady motion of the fluid 

 within a portion of a tube of flow bounded by cross-sections perpendicular to the 

 stream-lines. In this case the surface-integral on the right is limited to the ends of 

 the tube. Let dS l5 c/S 2 be the areas of the ends, and q v q 2 be the velocities there 

 along the tube. Then the surface integral becomes 



(hi + Vi + P A PifcdSx - (te\ + v 2 +^Ws 2 . 

 \ pj \ Pi/ 



But since the motion is steady, p^d^ = p 2 q 2 d$ 2 , since the former is the mass of fluid 



