THE EQUATION'S OF FLTJID-MOTIOX. 139 



on the surface f, is S-^^^j^'V'P,S,a=S/ii8/Zj2, and is con- 

 stant in all time. 



If we compare the form of cr here obtained with that 

 assumed by Helmholtz, we find that his form is not suffi- 

 ciently general, since he writes 2!, v?i + 2, Vfz + -5 V^j 

 = V<yj where w is a vector with the condition Sv<»=o. 

 That is, he includes two undetermined quantities instead 

 of three, and does not express in a distinct form the essen- 

 tial condition (namely D^S = o) affecting them. 



The quantities 2 are in no way dependent on finite 

 forces which are acting (under the hypothesis), but entirely 

 on the initial conditions (boundary conditions not now 

 being entertained) . 



Unless the density be a function of the pressure only, 

 the relations just proved will certainly not hold ; for then 



DfO- + vV + ^Vi^=o, 



and 



..11. ^^^+-„ 



where S,=/(ip,, <25,, (Pj, and /), in which case the fluid- 

 pressures will of themselves cause vortex motion. 



This seems to be a quite possible condition in a gas 

 rapidly heated or expanded. 



Now turning to the terms containing /i, and imagining d 

 a function oi p only, and remembering that /i is independ- 

 ent of c?, then 



SVo-=gD,H = ^D,</=D, log d 



