182 REPORT—1858. 
conceive the number of such steps to become infinitely great, we are led at 
once to the conception of the triangular, instead of the rectangular notch. 
The principle of the triangular notch being thus arrived at, it becomes 
evident that there is no necessity for having one side of the notch vertical 
and the other slanting; but that, as may in many cases prove more con- 
venient, both sides may be made slanting, and their slopes may be alike. It 
is then to be observed, that by the use of the triangular notch with proper 
formulas and coefficients derivable by due union of theory and experiments, 
quantities of running water, from the smallest to the greatest, may be accu- 
rately gauged by their flow through the same notch. The reason of this is 
obvious, from considering that in the triangular notch, when the quantity 
flowing is very small, the flow is confined to a small space admitting of 
accurate measurement ; and that the space for the flow of water increases as 
the quantity to be measured increases, but still continues such as to admit 
of accurate measurement. 
Further, the ordinary rectangular notch, when applied for the gauging of 
rivers, is subject to a serious objection from the difficulty, or impossibility, 
of properly taking into account the influence of the bottom of the river on 
the flow of the water to the notch. If it were practicable to dam up the 
river so deep that the water would flow through the notch as if coming from 
a reservoir of still water, the difficulty would not arise. This, however, can 
seldom be done in practice ; and although the bottom of the river may be so 
far below the crest as to produce but little effect on the flow of the water 
when the quantity flowing is small, yet when the quantity becomes great, 
the “ Velocity of Approach” comes to have a very material influence on the 
flow of the water, but an influence which it is usually difficult, if not im- 
practicable, to ascertain with satisfactory accuracy. In the notches now 
proposed, of triangular form, the influence of the bottom may be rendered 
definite, and such as to affect alike (or at least by some law that may be 
readily determined by experiments) the flow of the water when very small, 
or when very great, in the same uotch. The method by which I propose 
that this may be effected, consists in carrying out a floor starting exactly 
from the vertex of the notch, and extending both up stream and laterally so 
as to form a bottom to the channel of approach, which will both be smooth, 
and will serve as the lower bounding surface of a passage of approach un- 
changing in form while increasing in magnitude at the places at least which 
are adjacent to the vertex of the notch. The floor way either be perfectly 
level, or may consist of two planes whose intersection would start from the 
vertex of the notch, and, as seen in plan, would pass up stream perpendicularly 
to the direction of the weir board, the two planes slanting upwards from 
their intersection more gently than the sides of the notch. The level floor, 
although theoretically not quite so perfect as the floor of two planes, would 
probably, for most practical purposes, prove the more convenient arrangement. 
With reference to the use of the floor, it may be said, in short, that by a due 
arrangement of the notch and the floor, a discharge orifice and channel of 
approach may be produced, of which (the upper surface of the water being 
considered as the top of the channel and orifice) the form will be unchanged 
or but little changed, with variations of the quantity flowing ;—very much 
less certainly than is the case with rectangular notches. 
The laws regulating the quantities of water flowing in such orifices as 
have now been described, come naturally next to be considered. Without, 
however, in the present interim Report, attempting to enter on a detailed 
discussion of theoretical considerations on this subject, I shall here merely 
advert briefly to the principal results and methods of reasoning. 
