80 Mr Weatherburn, On the Hydrodynamics of Relativity 



Expanding the second term and using the equation of con- 

 tinuity, we find 



rfw T^ dk _ 



dt k dt ~ ' 



which, after division by k, may be written 



d fVT\ w ^ 



s(l)=I-^^ ■; (23). 



Differentiation with respect to t gives 



cZ- /w\ fd Mv\ -. w ;'d _ 



If then w vanishes at any instant it follows from (23) that the 

 first derivative of w/Z; also vanishes, and from the next equation 

 likewise the second derivative at that instant. Similarly all the 

 derivatives with respect to t vanish at that instant, and the 

 quantity -wjk remains permanently zero, so that the motion con- 

 tinues irrotationa.l. 



Further, the moment of a vortex filament does not vary with the 

 time. For if ds is an element of such a filament moving with the 

 fluid 



ds = wds/w, 



and -J- (ds) = ds* Vv = — w • Vv, 



at w 



so that (23) is equivalent to 



d /wX w d , -, ^ 



<sUJ = arf(('^'> (24). 



Now if jj, is the moment of the filament, dm^ the constant 

 normal rest-mass of the element considered, and a the cross- 

 sectional area 



fx = aw, dniQ = kads, 

 ,1 , w ds 

 ^°"'''* k = ^d^. (25)- 



Substituting this value in (24), and remembering that dm^ is 

 constant, we have 



|(/.^s) = ;x|(rfs), 



and therefore -^ = 0, 



dt 



showing that the moment of the filament remains constant. 



