THE EQUATIONS OF FLUID-MOTION. 135 



and 



HVVcr = V(«D,v<p,+/3D,Vfi + 7D<V'P3). • (8) 



Of these {7) gives the more convenient form, from 

 which Ave obtain 



HVvo-=Vv<PiD<Vv<P2V'Pj+ &c. 



= V V ^x VJD.V^z V<P3 + Vvf jVVc;^zD/Vf J + &c. ; 

 and applying to each of these expressions the formula 



we obtain 



==V<p,\Sv<pJ>tV<Pi-^V<PiDtV<Pz\-\-8cc. + kc., 

 whence 



SV<p.DiV<P; = Svf5DiV<P., &c., 

 if the motion is irrotational. Also expressing 



Vvcr = /,a + /./S + /37, 

 we find, by operating with SVfi on each side, 



- IP/, = S vf J V^.D,/3 + S vf , vf jD^Y 



therefore 

 and 



