127, 128.] PROPERTIES OF VORTEX-LINES. 149 



where the integration extends over any closed surface lying wholly 

 in the fluid. Applying this to the closed surface formed by two 

 cross-sections of a vortex-tube, and the portion of the tube in 

 tercepted between them, we find co^ = a 2 o- 2 , where o^, o&amp;gt; 2 denote 

 the angular velocities at the sections cr x , &amp;lt;r 2 , respectively. 



Thomson s proof shews that the theorem is true even when 

 f, 77, f are discontinuous (in which case there may be an abrupt 

 bend at some point of a vortex), provided only that u, v, w are 

 continuous. 



An important consequence of the above theorem is that a 

 vortex-line cannot begin or end at any point in the interior of 

 the fluid. Any vortex-lines which exist must either form closed 

 curves, or else traverse the fluid, beginning and ending on its 

 boundaries. Compare Art. 44. 



The theorem (6) of Art. 40 may now be enunciated as follows : 

 The circulation in any circuit is equal to twice the sum of the 

 strengths of all the vortices which it embraces. 



128. The motion of the fluid occupying any simply-connected 

 region is determinate when we know the values of the expansion 

 (6, say,), and of the component angular velocities f, 77, f at every 

 point of the region, and the value of the normal velocity (X, say,) 

 at every point of the boundary. 



If possible, let there be two sets of values, u lt v lt w l} and 

 u 2 , v z , w 2 , of the component velocities, each satisfying the above 

 conditions ; viz. each set satisfying the differential equations 



du dv dw _ fi ,. 



1 -- \&quot;T&quot;T ~T~ &quot; ..................... WJ 



ax ay dz 



dw dv _ fc du dw _ dv du_^ ... 



dy~dz~ ^ Tz~ dx--* 1 Tx~dy~ 



throughout the region, and the condition 



lu + mv + nw =X ....................... (5), 



at the boundary. Hence the quantities 



u = t/. x w t , v = v l r a , w = w l w t , 



