154 VORTEX MOTION. [CHAP. VI. 



gral in question will still vanish. On this understanding then 

 the relation (8) is satisfied, and the values of u, v, w thus derived 

 (which we shall now distinguish by a new suffix) will satisfy (3) 

 and (4). 



They will not however in general satisfy the boundary-condi 

 tion (5). Let X be the value of the normal velocity which the 

 formula (6) would give, viz. 



\ = ^ o + mv o + nw &amp;gt; 

 and let us write 



u = u 4 u lt v = v Q + v lt w = w + w lf 



where u lf v lt w : remain to be found. Substituting in (3), (4) and 

 (5) we obtain 



dx dy dz 



dw 1 _dv 1 = Q du i_^i == Q ^ y i_ c l!!L = o- 

 dy dz dz dx dx dy 



with the boundary condition 



lu t + mv 1 + nw^ = X X . 

 Hence we may write 



dQ dQ dQ 



ni - _ ;* n\ - _ ^L nil - _ 5 



U ^~dx&amp;gt; ^~dy i~ dz 



where Q is a single-valued function satisfying 



V 2 &amp;lt;?=0 ....................... ........ (17) 



throughout the (simply-connected) region, and making 



at the boundary. The problem of finding Q so as to satisfy these 

 conditions was shewn in Art. 49 to be determinate. 



Vortex-sheets. 



132. We have so far assumed u, v, w to be continuous. We 

 will now shew how cases where these functions are. discontinuous 

 may be brought within the scope of our theorems. 



Let us suppose that we have a series of vortex-filaments ar 

 ranged in a thin film over a surface S, and let w be the angular 



