260 REPORT— 1876. 



similar V-notches, we denote the height of the still-water surface-level above 

 the level of the vertex of the notch by 1i, we have 



Qi=cU, (9) 



where c is a constant coefficient, which cannot be determined by theory, but 

 can be very satisfactorily determined by experiment for any desired ratio of 

 horizontal width to vertical depth to be adopted for the form of the notch. 

 Experiments determining the values of c for certain forms and arrangements 

 of V-notches, suited for practical convenience and utility, have already been 

 made by myself, and have been reported on to the British Association ; and 

 the Reports on them are printed in the British Association volume for Leeds 

 Meeting, 1858, and in that for Manchester Meeting, 1861. 



Investigation of a Foemvia for the Flow of "Water in a Rectangular 

 Notch with Level Crest in a Vertical Plane Face. 



It is to be premised that the long-known and generally used formulas for 

 the flow of water in rectangular notches, brought out by the so-called 

 " theories " which I have dissented from in the earlier part of the present 

 paper, have been mainly of the form 



where Q, denotes the volume per unit of time, 



L denotes the horizontal length of the notch, 



h the vertical height from the crest of the notch to the stiU-water 



surface-level, and 

 g the coefficient for gravity, 

 and where c has either been taken as a constant numerical coefficient for 

 want of accurate experiments to determine its values for different values of 

 L and li, or has been treated as a variable. Poncelet and Lesbros have 

 taken this latter course, and have deduced by experiments extensive tables 

 of its values for different depths of water in notches of the width on which 

 they experimented — a width, namely, of 20 centimetres*. As, however, the 

 coefficient for terrestrial gravity varies but little for different parts of the 

 world, it has most frequently been left out of account, a single coefficient c! 

 being used instead of eg ; so that if, for instance, when the foot and second 

 are used as units of length and time, we take 32-2 as a correct enough state- 

 ment of the value of g for any part of the world, we have c' = 32-2c. 



A new formula, involving an important improvement in its form and 

 adjusted so as to be in due accordance with numerous elaborate experiments, 

 was developed within or about the time from 1846 to 1855, in America, by 

 Mr. Boyden and Mr. Francis, both of Massachussetts. It is 



Q=3-33(L-Jjj«70^% 



where Q is the quantity of water in cubic feet per second, 

 L is the length of the notch in feet, 



h is the height from the level of the crest to the still-water surface- 

 level in feet, and 

 n is the number of end contractions, and must be either 0, 1, or 2. 



* M^moires de 1' Acad^mie des Sciences : Sciences Math^matiques et Physiques, tome iii. 

 1829, 



