600 Prof. W. M. Hicks on the 



Therefore \x 2 vds taken round the stationary core boundaiy 



The corresponding theorem in two dimensions is 

 J xvds = \ix, 



where x is the distance of the centre of gravity of the section 

 from the translation axis. 



For region (3), the energy in the translation state is 



E 3 =E 3 + E 2 — E 2 



— 7r/i%2 — 2 (vol. core) U 2 -f \ ttJJxx 2 v ds — E 2 , 



the integral being taken over the core section 



= 7r/i,^ 1 + ^(total vol. carried forward) D 2 + ^fjuUk 2 * 



When the portion carried forward is singly-connected, %i = 0, 



E 3 = ^fjbTJk 2 — ^(translated mass) U 2 . 



This can be verified at once in the case of the spherical 

 vortex in which U = /i/5c, / L ,2 = §c 2 . This gives E 3 = ;|mU 2 or 

 energy of translation of half mass of fluid displaced by the 

 sphere, and is correct. 



The corresponding theorem for two dimensions is on one 

 side of axis of x 



E 3 = ^U^' — \ (translated mass) U 2 , 



or for the whole motion 



E 3 = //,U^ — \ (translated mass) U 2 , 



E 2 = //<%2 + i (mass of region 2) U 2 . 



Of this general theory the present note discusses in more 

 detail the two special cases of (a) two parallel straight 

 vortices, (b) a single ring vorte* of uniform voracity. 



(a) Two parallel straight vortices. 



Take first the case where they are so far apart that their 

 sections can be regarded as circles of radius c, with centres 

 at a distance 2a. If co denote the rotational constant and 

 Jul the circulation, ^ — 2irc . o)c = 27rc 2 a). The velocity at any 

 point due to one vortex, at a distance r from it, is <wc 2 /r = 

 fMJ(27Tr). In the figure A, A' are the centres of the vortices,. 



