Mr Weatherbarn, On the Hydrodijnaniics of Relativitij 79 



I shall now show that, on the assumption of a force potential, 

 Kelvin's theorem* of the constancy of the circulation in a closed 

 filament moving with the fluid is true in the present case also. 

 Consider a closed filament consisting always of the same particles, 

 and let ds be a vector element of its length and ds the correspond- 

 ing scalar. Then the circulation round it is 



/ = kv •ds. 



The time rate of change of this is 



dl 



dt ' 



|(.v).r/s+.v.(;jj 



f/s 



VF--VP + /cv.(f/s.V)v 



7 



--F+/CV. _- 



OS OS 



ds 



ds 



"lv" 



.(21). 



Now the last integral is 



ds 



c- c 



, V/c — 



(c"- 



^. die K 8y^ , , 



''hs'-^.Ts^^'- 



T 27 Vc2 - V- 



On substitution of this value in (21) that equation reduces to 



dr 



dt 



9«: 



9^ 9 / n' 



96' ds ds 



ds. 



Hence, since the path of integration is closed and k, V, and kv^ 

 are single-valued functions, the integral vanishes, showing that 



f- 



.(22). 



Thus the circulation does not alter with the tiipe. 



Corollary. If / is zero at any instant it will remain zero. In 

 particular, if the motion is irrotational at any instant it will remain 

 so, provided that the impressed forces have a potential. 



§ 9. Helmholtz's theoremsf. That these theorems are true in 

 the present theory also follows without difficulty from the form (8') 

 of the equation of motion. For taking the curl of both members 

 we have 



dw 



'dt 



+ curl (w X v) = 0. 



* Of. Silberstein, loc. cit. p. 161, for the proof of the ordinary theorem. 

 t Ibid. Y>p. 163 — 65. 



