MECHANICS AND USEFUL ARTS. 59 



of the theory described with the experiments ; because a table of experi- 

 mental coefficients for various depths, or an empirical formula slightly mod- 

 ified from the theoretical one, will serve all purposes. 



To one evident simplification in the proposed system of gauging, as com- 

 pared with that by rectangular notches, I would here advert, namely, that 

 in the proposed system the quantity flowing comes to be a function of only 

 one variable, namely, the measured head of water, while in the rectangular 

 notches it is a function of at least two variables, namely, the head of water 

 and the horizontal width of the notch, and is commonly, also, a function of 

 a third variable very difficult to be taken into account, namely, the depth 

 from the crest of the notch down to the bottom of the channel of approach ; 

 which depth must vary in its influence with all the varying ratios between 

 it and the other two quantities of which the flow is a function. 



The proposed system of gauging also gives facilities for taking another 

 element into account, which often arises in practice, namely, the influence 

 of back water on the flow of the water in the gauge, when, as frequently 

 occurs in rivers, it is found impracticable to dam the river up sufficiently to 

 give it a clear overfall free from the back or tail water. For any given ratio 

 of the height of the tail water above the vertex of the notch, I would antici- 

 pate that the quantities flowing would still be, approximately at least, pro- 

 portional to the power of the head as before, and a set of coefficients 

 would have to be determined experimentally for different ratios of the height 

 of the tail water above the vertex of the notch. 



"\Vitli the aid of the grant placed at my disposal by the Association at last 

 year's meeting, for the purpose of these researches, I have got an experi- 

 mental apparatus constructed and fitted up at a place a few miles distant 

 from Belfast, and I have got some preliminary experiments made on a right- 

 angled notch in a vertical plane surface, the sides of the notch making 

 angles of 4-3 with the horizon, and the flow being from a deep and wide 

 pool of quiet water, and the water thus approaching the notch uninfluenced 

 by any floor or bottom. The principal set of experiments as yet made were 

 on quantities of water varying from about two to ten cubic feet per minute, 

 and the depths or heads of the water varied from two to four inches in the 

 right-angled notch. 



From these experiments I derive the formula Q = 0'317 H |-, where Q is the 

 quantity of water in cubic feet per minute, and H the head as measured ver- 

 tically in inches from the still water level of the pool down to the vertex of 

 the notch. This formula is submitted at present temporarily, as being accu- 

 rate enough for use for ordinary practical purposes, for the measurement of 

 water by notches similar to the one experimented on, and for quantities of 

 water limited to nearly the same range as those in the experiments ; but as 

 being, of course, subject to amendment by more perfect experiments extend- 

 ing through a wider range of quantities of water. 



It will be readily observed that the experimental investigations indicated 

 in the foregoing report as desirable, are such as would require for their com- 

 pletion and extension to large flows of water a great expenditure botji of 

 time and money, like as has already been the case with researches on the 

 flow of water in rectangular notches. All that I can myself, for the present, 

 propose to attempt, is to open up the subject with experiments on moder- 

 ately small flows of water. 



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