—_—— = 
EXPERIMENTS ON THE MEASUREMENT OF WATER. 183 
By theory I have been led to anticipate that the quantity flowing ina 
given notch should be proportional, or very nearly so, to the 3 power of the 
lineal dimensions of the cross section of the issuing jet, or to the 3 power of 
the head of water over the vertex of the notch. This head is to be under- 
stood, in the case of water flowing from a still reservoir, as being measured 
vertically from the level of the water surface down to the vertex of the 
notch; or, in the case of water flowing to the notch, with a considerable 
velocity of approach over a floor arranged as above prescribed, the head is 
to be considered as being measured vertically from the water surface where 
the motion 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 enunciated, to the effect that the quantity flowing should he pro- 
portional to the 3 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 viscidity and friction, and free from capillary attraction at its 
surface, and from any other slight disturbing causes that may have minute 
influences 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 suffi- 
cient distance from the notch, while increasing in magnitude as the flow in- 
creases (such being supposed, according to my theory, to be possible) ; and 
if due attention be paid to measuring 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 sections of 
the two jets at the notches may be similarly divided into the same number 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 sections, or, 
as from various considerations may readily be assumed, proportional 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 considerations, 
supported by numerous others, it appears that the quantities Mowing should 
be proportional to the products of the squares of the heads into their square 
roots, or to the $ powers, 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 havea 
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 velo- 
cities, such as occur in notches used for practical purposes, unless unusually 
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, although slightly influencing the absolute 
amount of the flow, will not, according to that assumption, at all interfere 
with its proportionality to the 3 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 ; 
