G4 IRROTATIONAL MOTION. [CHAP. III. 



and the rest over the several barriers. The coefficient of any K is 

 evidently the total flux across the corresponding barrier, in a 

 motion of which -^ is the velocity-potential. The values of $ in 

 the first and last terms of the equation are to be assigned in the 

 manner indicated in Art. 56. 



If ty also be a cyclic function, having the cyclic constants 

 KI, # 2 , &c., then (22) becomes in the same way 



(31). 



Equations (30) and (31) together constitute Thomson s extension 

 of Green s theorem. 



67. If in (30) we put -fy = $, and suppose (j&amp;gt; to be the velocity- 

 potential of an incompressible fluid, we find 



To interpret the last member of this formula we must recur to the 

 artificial method of generating cyclic irrotational motion explained 

 in Art. 61. The first term has already been interpreted as twice 

 the work done by the impulsive pressure p(f&amp;gt; applied to every 

 part of the original boundary of the fluid. Again, p/c l 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 in Art. 61 supposed to be occupied ; so that the 



expression ^ 1 1 p/c : . -~ do- l denotes the work done by the impulsive 



forces applied to that membrane ; and so on. Hence (32) expresses 

 the fact that the energy of the motion is equal to the work done 

 by the whole system of impulsive forces by which we may suppose 

 it generated. 



