42-43] GREEN'S THEOREM. 49 



Hence (1) simply expresses the fact that the surface-integral (2), 

 taken over the boundary of the region, is equal to the sum of the 

 similar integrals taken over the boundaries of the elementary 

 spaces of which we have supposed it built up. 



The interpretation of this result when U, V, W denote the 

 component velocities of a continuous substance is obvious. In the 

 particular case of irrotational motion we obtain 



where Bn denotes an element of the inwardly-directed normal to ^ 

 the surface S, *-5,?5 



Again, if we put U, F, W = pu, pv, pw, respectively, we 

 reproduce in substance the investigation of Art. 8. 



Another useful result is obtained by putting U, F, W 

 U(f>, V(j>, W(f), respectively, where u, v, w satisfy the relation 



du dv dw _ - 



doc dy dz ~ 

 throughout the region, and make 



lu + mv + nw = 

 over the boundary. We find 



The function <j!> is here merely restricted to be finite, single- valued, 

 and continuous, and to have its first differential coefficients finite, 

 throughout the region. 



43. Now let <f), <f>' be any two functions which, together with 

 their first derivatives, are finite, continuous, and single-valued 

 throughout the region considered ; and let us put 



respectively, so that 

 Substituting in (1) we find 



^f*Sf>^ 



(5) 



