Waves in Flowing Water. 447 



This, and the corresponding expression relatively to A , 

 gives, by (3), the sum of horizontal pressure on all inequalities 

 between A and A, when the problem of the fluid motion in 

 the circumstances is so far solved as to give D, C\, and w 2 — v 2 

 for each of the end sections A , A. 



Suppose, now, A to be so far on the up-stream side of the 

 inequalities that the motion of the water across it is sensibly 

 uniform and horizontal, with velocity which we shall denote by 

 U ; so that, for A , (6) becomes 



{x + J^tAfyJ^fcDo'+Uo'Do .... (7). 



Hence, and by (6) and (4), 

 F=^(D„ 2 -D 2 ) + U 2 D -iq 2 D-|JJ M 2 - l , 2 )^ . (8). 



Now, by the law of velocity at the free surface in steady 

 motion, we have 



W = iW+gCD -D) .... (9); 



because, the points B , B of the bottom being on the same 

 level, D — D is the difference of levels between the surface- 

 points ty and ty. Hence (8) becomes 



F = ^(D -D) 2 + Uo 2 (D -D)-KU 2 -U 2 )D 



+ ifV+u 2 -w 2 )% • (10), 



where U denotes a constant which may have any value. It is 

 convenient to make it the mean horizontal component velocity 

 across $B : we therefore take 



U 



i r D 



= dJ/^ < u ) 



and, because the quantities flowing in across A and out across 

 A are equal, as the motion is steady, we have 



UD=U D (12). 



Using this to eliminate U from (10), we find 



F=i(y-^)(D -D) 2 +iJV + U 2 -» 2 )<% . (13). 



To evaluate D — D when we know enough about the mo- 

 tion, and to see how its value is related to other characteristic 

 quantities, let us look back to (9), and in it take 



q a = TP + t> a (14). 



