where "V" denotes the control volume. This is not the usual application of 

 the Lagally theorem, which gives the force on a closed body. Here it 

 represents the reaction on the Betz sources due to the flux of momentum of 

 the source-generated discharge. 



Since the flux through the bounding surface T is the same with Betz 

 sources as in the solenoidal (divergenceless) real flow, the difference in 

 flux for the flows through the area A of S intersected by the wake is at- 

 tributable to the Betz sources. The Gauss flux theorem then yields the 

 formula 



4lT 



y dx = (u^-u) dS (73) 



Far downstream, the section S will be denoted by S^ and the wake area 

 by A . In terms of the flux Q across this wake area, we have the well- 

 known formula for the drag. 



-J- 



D = p U^Q, Q = - u dS (74) 



oo 



A similar expression for the force on the Betz sources, obtained from the 

 asjmiptotic form of (71) , is 



"o/ "1 



D„ = - p U^ u, dS 



But 



dS 

 'S , 'S "S, 



CO* CO I 



u^ dS = (u^-u) dS + u 



and since u^ = u except within the wake and, by continuity, I u dS = 0, 



we obtain S 



29 



