156 REPORT— 1861. 



increases somewhat more rapidly than the width of the notch for a given depth. 

 Now, it is to be observed that the contraction of the stream issuing from an 

 orifice open above in a vertical plate is of two distinct kinds at different parts 

 round the surface of the vein. One of these kinds is the contraction at the 

 places where the water shoots off from the edges of the plate. The curved 

 surface of the fluid leaving the plate is necessarily tangential with the surface 

 of the plate along which the water has been flowing, as an infinite force 

 would be required to divert any moving particle suddenly out of its previous 

 course*. The other kind of contraction in orifices open above consists in 

 the sinking of the upper surface, which begins gradually within the pond or 

 reservoir, and continues after the water has passed the orifice. These two 

 contractions come into play in very difl'erent degrees, according as the notch 

 (whether triangular, rectangular, or with curved edges) is made deep and 

 narrow, or wide and shallow. From considerations of the kind here briefly 

 touched upon, I would not be disposed to expect theoretically that the coeffi- 

 cient c for the formula for Y-shaped notches should be at all truly proportional 

 to the horizontal width of the orifice for a given depth ; and the experi- 

 mental results last referred to are in accordance with this supposition. I 

 would, however, think that, from the experimental determination now arrived 

 at, of the coefficient for a notch so wide as four times its depth, we might 

 very safely, or without danger of falling into important error, pass on to 

 notches wider in any degree, by simply increasing the coefficient in the same 

 ratio as the width of the notch for a given depth is increased. 



Appendix. — April 1862. 



With reference to the comparison made, in the concluding sentences of the 

 foregoing Report, between tiie quantities of water which, for any given depth 

 of flow, are discharged by notches of different widths, and to the opinion 

 there expressed, that we might, without danger of falling into important 

 error, pass from the experimental determination of the coefficient for a 

 notch so wide as four times its depth, to the employment of notches wider in 

 any degree, by simply increasing the coefficient in the same ratio as the width 

 of the notch for a given depth is increased, I now wish to add an investi- 

 gation since made, which confirms that opinion, and extends the determina- 

 tion of the discharge, beyond the notches experimented on, to notches of any 

 widths great in proportion to their depths. This investigation is founded on 

 the formula for the flow of water in rectangular notches obtained from ela- 

 borate and careful experiments made on a very large scale by Mr. James B. 

 Francis, in his capacity as engineer to the Water-power Corporations at 

 Lowell, Massachusetts, and described in a work by him, entitled ' Lowell 

 Hydraulic Experiments,' Boston, }855f. That formula, for either the case 

 in which there are no end-contractions of the vein, or for that in which the 

 length of the weir is great in proportion to the depth of the water over its 

 crest, and the flow over a portion of its length not extending to either end is 

 alone considered, is 



Qj=3-33L,H^t (1) 



where Lj = length of the weir over which the water flows, without end-con- 

 tractions; or length of any part of the weir not extending to 

 the ends, in feet : 



* This condition appears not to have been generally noticed by experimenters and writera 

 on hydrodynamics. Even MM. Poncelet and Lesbros, in their dehncations of the forms 

 of veins of water issuing from orifices in thin jolates, after elaborate measurements of those 

 forms, represent the surface of the fluid as making a shai-p angle with the plate in leaving 

 its edge. f The formida is to be found at page 133 of that work. 



