THE EQUATIONS OF FLUID-MOTION. 133 



we see that 



symbolically, where A is the same operator as y, but acts 

 on the 0- only. 

 Thus 



D^V^ = yD,<p + A (So-vf) = VD/<P + VfxSy^Vf 



+ W<PS^^J<p + Vh^'^^Vh .... (4) 

 since 



[These other two forms of v will also be used, 



or 



= «S vf . V + /3S vf .V + 7S<P3 V. 

 since y is in form a vector.] 



Let us first apply these results to SvPiVPiVPj or — H ; 



therefore 





or 



^D,H-Svcr=-HD,(g), (5) 



or is evidently rate of compression per unit of volume. 

 If, again, we act similarly on a or V\;(p^^<p , 



