184 REPORT—1858. 
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 of the theory described with the experiments, because in 
respect to any given form of notch and approach, a table of experimental coefh- 
cients for various depths, or an empirical formula slightly modified from — 
the theoretical one, will serve all purposes. ‘To one evident simplification 
in the proposed system of gauging, as compared with that by rectangular 
notches, I would here advert, namely, that in the proposed system, when 
once the form of the notch and channel of approach is fixed for gauging 
any set of streams, the quantity flowing comes to be treated as a function 
of only one variable, namely, the measured head of water; while in the rect- 
angular notches it is practically treated as a function of at least two variables, 
namely, the head of water and the horizontal width of the notch; because 
in practice it would be inconvenient, if not impossible, to select any single 
width of notch, or any moderate number of widths of notches, for general 
use, for very varied quantities of water. It 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, to the 
height of the head water above the vertex of the notch, I would anticipate 
that the quantities flowing would still be, approximately at least, proportional 
to the 5-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 head 
water to the height of the tail water above the vertex of the notch. 
With the aid of the grant placed at my disposal by the Association at last 
year’s meeting, for the purposes of these researches, I have got an experi- 
mental apparatus constructed and fitted up at a place a few miles distant 
from Belfast, in Carr’s Glen, on the grounds of Mr. Neeson, who has kindly 
afforded me all the necessary facilities regarding the water supply and the site 
forthe experiments ; 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 45° 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 2 to 10 cubic feet per minute ; 
and the depths or heads of water varied from 2 to 4 inches in the right- 
angled notch. From these experiments I derive the formula Q="317 H?; 
where Q is the quantity of water in cubic feet per minute, and H the head 
as measured, vertically, in inches, from the still-water level of the pool, down 
to the vertex of the notch. ‘This formula is submitted, at present tempo- 
rarily, as being accurate enough for use for ordinary practical purposes for 
