ON THE FLOW OF WATER THROUGH ORIFICES. 261 



This formula was ofFcrcd by Mr. James B. Francis, in his work entitled 

 " Lowell Hj-draiilic Experiments," and published at Boston in 1855, not as 

 one founded on any complete theoretical views, but as one depending on 

 several assumptions probably not perfectly correct, and yet as one which, 

 through numerous trials and by adjustments introduced tentatively in fitting 

 it to experimental results, had been brought out so as to agree very closely 

 with experiments. 



In § 120, at page 72 of his work, Mr. Francis says : — " No correct formula 

 " for the discharge of water over weirs, founded upon natural laws, and in- 

 " eluding- the secondary eflects of these laws, being known, we must rely 

 " entirely upon experiments, taking due care in the application of any formula 

 " deduced from thence not to depart too far from the limits of the experiments 

 " on which it is founded." And in §§ 123, 124, at page 74, in respect to the 

 conception of the formula, he further gives the following very clear expla- 

 nations : — " The contraction which takes place at the ends of a weir dimi- 

 " nishes the discharge. When the weir is of considerable length in proportion 

 " to the depth of the water flowing over, this diminution is evidently a con- 

 " stant quantity, whatever may be the length, provided the depth is the same ; 

 " we may, therefore, assume that the end contraction effectively diminishes the 

 " length of such weirs, by a quantity depending only upon the depth upon 

 " the weir. It is evident that the amount of (his diminution must increase 

 " with the depth ; we are unable, however, in the present state of science, to 

 " discover the law of its variation ; but experiment has proved that it is very 

 " nearly in direct proportion to the depth. As it is of great importance, in 

 " practical applications, to have the formula as simple as possible, it is assumed 

 " in this work [ilr. Francis's book] that the quantity to be subtracted from 

 " the absolute length of a weir having complete contraction, to give its effective 

 " length, is directly proportional to the depth. It is also assumed that the 

 " quantity discharged by weirs of equal effective lengths varies according to a 

 " constant power of the depth. There is uo reason to think that either of 

 " these assumptions is perfectly correct ; it will be seen, however, that they 

 " lead to results agreeing very closely with experiment. 



" The formula proposed for weirs of considerable length in proportion to 

 " the depth upon them, and having complete contraction, is 



"Q = C(L-6n7i)7t*; 



" in which Q, = the quantity discharged in cubic feet per second ; 



" C = a constant coefficient ; 



" L=the total length of the weir in feet ; 



" 6 = a constant coefficient ; 



" n = the number of end contractions. In a single weir having com- 

 plete contraction, n always equals 2 ; and when the length 

 of the weir is equal to the width of the canal leading to it, 

 n = 0; 



"A = the depth of water flowing over the weir taken far enough 

 upstream from the weir to be unaffected by the curvature 

 in the surface caused by the discharge ; 



"a=a constant power." 



This formula, Mr. Francis states, was first suggested to him by Mr. Boyden 

 in 1846. 



The important novel feature in this formula consists in the subtraction 

 which it makes, from the length L of the notch, of a length for each end 



