Mr Weatherhurn, On the Hydrodynamics of Relativity 77 



This form is not so short as in the ordinary theory, nor can we 

 obtain Laplace's equation, as there, by assuming the Huid incom- 

 pressible, for such an assumption is inconsistent with the theory 

 of relativity*. 



§ 6. Steadily rotating fluid. Suppose that the Huid is in 

 a state of steady rotation about the ^^-axis, and that the angular 

 velocity of rotation O is a function of the distance r from that 

 axis. We shall now determine what must be the form of this 

 function in order that a velocity potential may existf- If i, J: k 

 are unit vectors in the directions of the coordinate axes 



V = rn. 

 For irrotational motion this velocity must satisfy the equation 



curl {kv) = 0, 



that is IkH + r V ('<:^) = 0, 



dr 



the integral of which is 



/tfir- = const. = fjb, 



say, so that /cH — ^ (A). 



The velocity potential <^ is then given by 



dd) . 1 d<h it 



dr r do r 



showing that (f) = fid + const (B), 



which is an example of a cyclic velocity potential. The integral 

 of the equation of motion is by (18') 



c"k + V=0 (C). 



But K involves v" and therefore O, which is itself expressed in 

 terms of k by (A). This equation however gives 



K' K-C^ 



whence 12- = -„ , „ „ ,„ , 



r-{/jb' + r''c-K-) 



* Cf. § 10 below. It will be shown, howeyer, in § 11 that V-</) = is the 

 equation of continuity for the steady irrotational motion of a Huid of minimum 

 compressibility. 



t Cf. Lamb, Hydrodynamics, § 28 (1st ed.). 



