NOTES ON HYDRODYNAMICS, CHIEFLY ON VORTEX-MOTION. 11 



The first two terms on the right give twice the rate of change of the surface 

 integral of elemental rotation (for any closed path of which AB forms part) due to 

 the motion and the lengthening of the elements of AB ; the last gives twice the 

 rate of change of this integral when the circuit is without motion. 



18. If we integrate round the closed path, then under the conditions already 

 stated as the dependence of p upon p, and the existence of the potential V, we get 

 from the vanishing of the left-hand side 



J2q^ sin eds +2 fq%h'+ f%ds =0 . . . (24), 



.' J os J dt 



and we see that the surface-integral of elemental rotation remains constant as the 

 closed path moves with the fluid. 



It is to be noticed that for any closed path moving with the fluid 



finds' = - / 2^ sin 6d*' .... (24'). 

 Equation (23) with the value of x inserted is 



Vb + / 7 - K - ( Va + / 7 - K) = ~ 2 / qojssS[n eds ' - 2 j qd S ds ' - !% ds ' ■ (25) ' 



B A AB 



and is equivalent to Lord Kelvin's theorem. The process of investigation here em- 

 ployed, though longer than that usually adopted, has the advantage of tracing the 

 various causes of change of the line-integral of flow along the path AB, and of throw- 

 ing some additional light on the meaning of the theorem. 



19. If we take AB parallel to the axis of x, we obtain 



J 2q<o sll > sin 6ds' = - f (2v£ - 2 M r))dx . . . (26). 



AB AB 



Let us now consider a fluid in motion in virtue of vortices contained within it, and 

 suppose that a surface S can be described within the fluid so that it is everywhere at 

 a finite distance from the vortices. We can describe a closed path, consisting of two 

 parts : (1) a part parallel to the axis of x, starting from a point A near the surface S, 

 and ending at a point B, also near the surface, but remote from A ; and (2) a part 

 starting from B and returning to A, but kept so near the surface everywhere as to 

 avoid the vortices. If this path move with the fluid, then by (24/) 



/ 4t<1x+ I 2qo> sx sin 6 sx dx + \ ~ds' + J 2qu>$ s > sin Ods = . . (27). 



AB AB BA BA 



This equation can be written 



J d "dx - f (2v£ - 2wT))dx + f d -^dx + f 2qo> sli > sin 6ds' = . . (27'). 



AB AB BA BA 



But going back to (21), we see that the last two terms on the left are X B — X A , and 

 as at every point of this part of the closed path &v = 0, we have 



X -X 



B A 



=/!*' • w 



BA 



