62 IRROTATIONAL MOTION. [CHAP. Ill 



value of Kp, and the values of the velocity-potential will therefore 

 differ by the corresponding value of tc ; finally, the motion on one 

 side of a barrier will be continuous with that on the other. To 

 prove the last statement we remark, first, that the velocities 

 normal to the barrier at two adjacent points on opposite sides of it 

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

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

 tive points on a barrier, and if the corresponding values of &amp;lt;/&amp;gt; be on 

 the positive side &amp;lt;p p , &amp;lt;f) Q , and on the negative side &amp;lt;/&amp;gt; p , &amp;lt;j&amp;gt; qt we have 



and therefore &amp;lt;f&amp;gt; Q &amp;lt;f&amp;gt; p = &amp;lt;/&amp;gt; Q &amp;lt;/&amp;gt; ,,, 



. *., if PQ = &, d&amp;lt;j)/ds = dp/ds. 



Hence the tangential velocities at two adjacent points on 

 opposite sides of the barrier also agree. If then we suppose the 

 barrier-membranes to be liquefied immediately after the impulse, 

 we obtain the irrotational motion in question. 



The physical interpretation of (4), when multiplied by p, 

 now follows as in Art. 44. The values of p/c are additional com 

 ponents of momentum, and those of ffd(j)/dn. dcr, the fluxes 

 through the various apertures of the region, are the corresponding 

 generalized velocities. 



55. If in (2) we put &amp;lt;f&amp;gt; &amp;lt;f&amp;gt;, and suppose &amp;lt; to be the velocity- 

 potential of an incompressible fluid, we find 



The last member of this formula has a simple interpretation in terms 

 of the artificial method of generating cyclic irrotational motion just 

 explained. The first term has already been recognized as equal 

 to twice the work done by the impulsive pressure p(f) applied to 

 every part of the original boundary of the fluid. Again, p^ is the 

 impulsive pressure applied, in the positive direction, to the in 

 finitely thin massless membrane by which the place of the first 

 barrier was supposed to be occupied ; so that the expression 



dA. 



-^ rfo-, 



-*// 



