264 REPORT — 1876. 



since the width has in general a constant ratio to the depth, or, in the case 

 more particularly considered, since the width is equal to the depth, the 

 quantity flowing per unit of time will, as in the preceding case, be propor- 

 tional to the I power of the depth ; or we have 



Mow over GK=j3h^^h, where fi is constant. 



Hence if q^ be the flow, in units of volume per unit of time, over a unit of 

 length in E F, we have 



q^=l3JWh. 



By multiplying this by I we get the quantity flowing over the entire middle 

 part E F per unit of time ; and so, denoting that quantity by Q", we have 



Q"=l3lh^/h, -^ 



or Q"=(i(L-mh)h'^hJ 



(11) 



Adding the expressions for Q' and Q," together, we get for the total flow in 

 the whole notch, which we may denote by Q, 



Q=p(L—mh)]WJi + aJiWh, 



or Q=l3LhVh-(i3m-a)JiWh, 



(3m — t 



or 



a=/3(L-fc^A);.l 



But — - — is a constant ; and let it be denoted by 26 ; and instead of the 



P 

 constant (i we may, in order now to use English letters, put a. Then 



Q=a(L-2bh)hi, (12) 



which is the desired formiila for the flow of water in a rectangular notcli 

 with two end contractions. 



This formula admits of easy modification to give a formula suitable for a 

 notch with only one end contraction*, thus: — 



Let the width of the notch with only oue end contraction be denoted by L 

 (as in fig. 14). Then conceive a notch twice as wide with two end contractions 

 as shown in fig. 15. The flow in this double space will, by the formula last 

 obtained (12), be seen to be =a(2L — 2bh)h^ ; and so if we put now Q to 

 deuote the flow in the notch under consideration (shown in fig. 14), which 

 will be half the flow in fig. 15, we have for the notch with only one end 

 contraction 



Q,=a{L-bh)h'- (13) 



* It is to be understood that contraction may be presented nt either end of a notch by 

 there being a vertical plane side face for the channel of approach to the notch, tliat side 

 face being perpendicular to the plane of the notch, and extending up-stream from the 

 notch so as to reach beyond the region of incipient rapid flow to the notch, and extending 

 for a little way down-stream past the notch, so as to afford the necessary guidance to the 

 issuing stream-filaments. In like manner, by two parallel vertical side walls or side faces 

 to the channel, when the crest of the notch extends quite across from the one wall-face to 

 the other, contraction may be prevented at both ends. 



