58 ANNUAL OF SCIENTIFIC DISCOVERY. 



is nearly stopped by the weir-board at a place near the board, but as far as 

 may be found practicable, from the centre of the notch. The law here enun- 

 ciated, to the effect that the quantity flowing should be proportional to the 

 power of the head, I consider should hold good rigidly in reference to 

 water flowing by a triangular notch in a thin vertical plate, from, a large and 

 deep reservoir of still water, if the water were a perfect fluid, free from vis- 

 cidity and friction, and from capillary attraction at its surface, and from any 

 other slight disturbing causes that may have minute influence on the flow, 

 the flow being supposed to be that due simply to gravitation resisted by the; 

 inertia of the fluid. The like may be said of water flowing from triangular 

 notches with shallow channels of approach, having floors as described above, 

 when due attention is given to make the passages of approach so as really 

 to remain unchanged in form for a sufficient distance from the notch, while 

 increasing in magnitude as the flow increases (such being supposed accord- 

 ing to my theory to be possible), and if due attention be paid to the measur- 

 ing the heads in all cases in positions similarly situated with reference to the 

 varying dimensions of the issuing streams. 



In illustration of these statements, or suppositions, I would merely say, 

 that, if two triangular notches, similar in form, have water flowing in them 

 at different depths, but with similar passages of approach, the cross section 

 of the two jets at the notches may be similarly divided into the same num- 

 ber of elements of area ; and that the areas of the corresponding elements 

 will be proportional to the squares of the lineal dimensions of the cross sec- 

 tions; or, as from various considerations may readily be assumed, propor- 

 tional to the squares of the heads; also the velocities of the water in the 

 corresponding elements may be taken as proportional to the square roots of 

 the lineal dimensions, or to the square roots of the heads. From these con- 

 siderations, supported by numerous others, it appears that the quantities 

 flowing should be proportional to the products of the squares of the heads 

 into their square roots, or to the power as already stated. 



The friction of the fluid on the solid bounding surfaces of the passages of 

 approach, where the water moves rapidly adjacent to the notch, may readily 

 be assumed, from all previous experience in similar subjects, not to have a 

 very important influence even on the absolute amount of the flow of the 

 water; and if we assume (as is known to be nearly the case for high veloci- 

 ties, such as occur in notches used for practical purposes, unless usually 

 small) that the tangential force of friction of the fluid per unit of area of 

 surface flowed along, is proportional to the square of the velocity of flow, it 

 follows by theory that the friction, though slightly influencing the absolute 

 amount of the flow, will not, according to that assumption, at all interfere 

 with its proportionality to the f power of the head. And this condition will 

 very nearly hold good if the assumption is very nearly correct. 



How closely the theory thus briefly sketched may be found to agree with 

 the actual flow of water, will be a subject for experimental investigation; 

 and whatever may be the result in this respect, the main object must be to 

 obtain for a moderate number of triangular notches of different forms, and 

 both with and without floors at the passage of approach, the necessary 

 coefficients for the various forms of notches and approaches selected, and for 

 various depths in any one of them, so as to allow of water being gauged for 

 practical purposes when in future convenient, by means of similarly formed 

 notches and approaches. The utility of the proposed system of gauging, it 

 is to be particularly observed, will not depend on a perfectly close agreement 



