82 Mr Weatherburn, On the Hydrodijnaribics of Relativity 



in virtue of the equation of continuit}-. On substitution of this 

 value in the last equation it becomes simply 



7/1- C" V" U/C Wn __ //-»/,N 



//divv+ ^ ~- = --^F.v (29), 



c- ot C^J 



which is the energy equation in terms of k' and v. These equations 

 (28) and (29) are identical with those found otherwise by Lamia* 

 and Lauef. The equation of continuity is as before 



^" + ^divv-0 (30), 



which takes the required form if k is replaced by jk'. 



§ 11. Steady irrotational motion. In virtue of (27) the 

 equation of continuity may also be written 



— + div(/cv) = (81), 



Ob 



and therefore when the motion is irrotational 



| + ^^^ = (31'). 



If it is also steady the iirst term is zero, and we have (as in the 

 older theory for the case of an incompressible fluid) 



^^(/) = (31"). 



Thus for steady irrotational motion of a fluid of niinimimi com- 

 pressibility the velocity jjotential satisfies Laplace s equation. 



It follows immediately that for such a fluid, filling a simply- 

 connected region within a hollow shell, which is fixed relative to 

 some system of reference S, steady irrotational motion relative to 

 that system is impossible. For by Green's theorem 



.K'V'dT = I {^(py^dr = — (f)/<:v • ndS — I (f)V-(f)dT. 



Now the last integral vanishes by the equation of continuity. 

 So also does the last but one : for v • n is zero, being the normal 

 velocity at the surface of the fluid. Hence 



/ 



fc-v'-dr — 0, 

 showing that v must vanish identically throughout the fluid. 



* Loc. cit. p. 792. 

 • t Loc. cit. p. 244. 



