ON THE GAUGING OP WATER BY TRIANGULAR NOTCHES. IS? 



Hi= height of the surface-level of the impounded water, measured 

 vertically from the crest of the weir, in feet : 

 and Q^= discharge in cubic feet per second over the length Lj of the weir. 



It is to be understood that, in cases to which this formula is applicable, 

 the weir has a vertical face on the upstream side, terminating at top in a 

 level crest ; and the water, on leaving the crest, is discharged through the 

 air, as if the weir were a vertical thin plate. 



To apply this to the case of a very wide triangular notch : — Let A B C be 



the crest of the notch, and A C the water level in the impounded pool. Let 

 the slopes of the crest be each m horizontal to 1 vertical ; or, what is the 

 same, let the cotangent of the inclination of each side of the crest to the 



horizon be =m. Let A E, a variable length, =^x. Then E D^ — . Let 



m 



E G be an infinitely small element of (he horizontal length or width from A 



to C. Then EG may be denoted h^ dx. Let ^-^quantity in cubic feet 



per second flowing under the length x, that is, under A E in the figure. 



Then dq will be the quantity discharged per second between E D and G F. 



Then, by the Lowell formula just cited, we have 



whence, by integrating, we get 



5r=3-33Ar-f^* + C, 

 mi 



in which the constant quantity is to be put ^0, because when a;=0, q also 

 =0. Hence we have 



9=fx3-33-L.a?^ (2) 



Let now H2= height in feet from the vertex of the notch up to the level 

 surface of the impounded water =BK in the figure. Then A K=m Hg. 

 Let also Qo = the discharge per second in the whole triangular notch = 

 twice the quantity discharged under A K. Then, by formula (2), we get 



Q,=|X3-33X^(«IH,)% 



or 



Qj=2-664 m H,* (3) 



To bring the notation to correspond with that used in the foregoing Report, 

 let Q=the quantity of water in cubic feet per minute, and H=the height 

 of the water level above the vertex in inches. 



Q H 



Then 0,^= and Yi^=-— ; and, by substitution in (3), we get 



Q=-320 m H* (4) 



This formula then gives, deduced from the Lowell formula, the flow in 

 cubic feet per minute through a very wide notch in a vertical thin plate, when 

 H is the height from the vertex of the notch up to the water level, in inches, 

 and when the slopes of the notch are each m horizontal to 1 vertical. 



