Stationary Waves in Water. 103 



Similarly on the lower Free surface close to the edge 



? 2 = ^{G(0)p, (5) 



- «<»)=(-';'"")' m 



where G is the value of G which will give the lower stream 

 surface. 



Hence, if y be measured at the point on the upper surface 

 where <£ = 0, 



G(iQ)-G(0)=f ^ [( ^ + H-y) 1 - {t + h) 1 ] , 



... (7) 



i.e. G0'Q)-G(O) = K, (8) 



where K is the right-hand side of (7). 



Evidently Q depends on the form of the function G(it'), 

 i. e. on the complete rigid boundary conditions both up and 

 down stream, and not merely on the state of affairs at the 

 edge and surface. 



As a first approximation, however, it may be assumed that 

 the function Gc(w) is dominated by the terms giving its 

 development near the point w? = corresponding to the edge 

 of the weir. 



Consider now 



-S = [ {i - G " w!!+iQ ' (tt) ]' • • (9) - 



Under the conditions postulated as to Q'(iv), this gives a 

 flow with a free surface along which q = l and over which 

 y=G(to)+C. 



Moreover, G(iQ) — G(O) is the ?/-component distance 



between the points <f> = on the stream-lines yjr = 0, ^ = Q 

 in the associated flow (9i. 



For example, take G(W) = e W7rQ , 



G(/Q) - G(O) = — Q= -6 11 Q, 



IT 



or if G(O) be assumed = G(0), (8) gives 



q= 3 - 27 [( h+ 0-( h+ £-^) 3 ] • (10) 



for flow over such a weir. 



