Mass carried forward by a Vortex. 599 



But we might have applied the general theory direct to the 

 actual motion. In this case, omitting the rotational region 

 for the moment, E = — ir^v'ds, in which the stream function 

 yjr is % + iUa? 2 , and v', the velocit}- along the boundary, is 



v + U -j- where v is the corresponding velocity in the 



stationary case. Hence 



E=-7rj( % + iU^)^ 5 -7rUj( % + iU^)^. 



Here \xdi/ = %\di/ = Q along a closed curve. 



\ x 2 dy along the two boundaries = (vol. of region). 



.*. E = E' + i (vol. of region) IP — \ir\] [a?vds. 



Comparing with the previous result it follows that 

 \x 2 vds--0, or that \x 2 vds round one boundary is equal to that 

 rouud the other. Clearly this theorem is general, and 

 states * that in a stationary condition \x 2 vds is the same along 

 any two stream lines, provided no rotational region intervenes 

 between them. 



For the core, the term 2Tr 2 fim~ 1 \\xtydxdy becomes 



^£ \ ( x X dxdy+ 7 ^ ; - \xHxdy. 



Now 27r\x d dxdy = [x 2 . 2-nxdxdy is the moment of inertia of 

 the whole ring round the translation axis =mk 2 , where k is 

 the corresponding radius of gyration, and is directly ex- 

 pressible in terms of the shape of the section. Thus 



E 1 = E 1 ' + £»iU 3 + IfMTJ/r-iTrU^rvds 



^Ei' + imU 2 . 



* The following direct proof of this, I owe to the kindness of Mr. G. 

 H. Livens. Let p, q denote the component velocities at a point on a 

 •stream line. Then integrating along two boundaries 



^v'cds= ^vHpd.v-t-q <?;,)= ff j — (x 2 q)~ - (x-p) [ dxdy 



= 2 xqdxdy 



= X quantity carried forward in the stationary state 



77 



=0. 



2 T 2 



