56 IRROTATIONAL MOTION. [CHAP. III. 



are the same, being each equal to the normal velocity of the 

 adjacent portion of the membrane. Again, if P, Q be two con 

 secutive points on a barrier, and if the corresponding values of &amp;lt; 

 be on one side &amp;lt;J)p, &amp;lt;Q, and on the other &amp;lt;j&amp;gt; p, &amp;lt;/&amp;gt; Q , we have, by (c) 



&amp;lt;j)p (j) p = K (j)Q (/&amp;gt; Q, 



or &amp;lt;&amp;gt; (&amp;gt;p = &amp;lt;&amp;gt; &amp;lt;&amp;gt; P, 



ds~ ds 



Hence the tangential velocities at two adjacent points on opposite 

 sides of a barrier also agree. If then we suppose the barrier- 

 membranes to be liquefied immediately after the impulse, we 

 obtain a state of irrotational motion satisfying the conditions 

 stated at the head of this article*. 



62. It is easy to shew analytically that the said conditions 

 completely determine (/&amp;gt;, save as to an additive constant. For, if 

 possible, let there be two functions fa, fa each satisfying the 

 conditions. Since &amp;lt;f) v fa have the same cyclic constants, 4 &amp;gt; = fa fa 

 is a single-valued function, which moreover satisfies (10) through 



out the region, and makes -7 = at every point of the boundary. 



Cv/c- 



Hence Art. 47 (/5) applies, and shews that &amp;lt;/&amp;gt; is constant. 



Hence the irrotational motion throughout an n -f- 1 -ply-con 

 nected space is determinate when we know the value of the normal 

 velocity at every point of the boundary, and also the value of the 

 circulation in each of the n independent circuits which can be 

 drawn in that space. 



The following theorem, which&quot; now replaces that of Art. 52, is 

 proved in like manner. 



The irrotational motion through an n+ 1-ply-connected region 

 extending to infinity, but limited internally by one or more closed 

 surfaces, is made fully determinate by the following conditions : 



* The modifications necessary in theorems (a) and (7) of Art. 48 are passed 

 over, as of little interest in our present subject. 



