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Proceedings of the Royal Society of Edinburgh. [Sess. 
charge to theoretical discharge. The numerical value of this coefficient 
varies somewhat irregularly, but not more so than could be expected with 
the somewhat imperfect apparatus used in the experiments. For all heads 
less than D (2*5 inches), excluding the first reading, the average coefficient 
is 0626. No measurements could be made for heads below T23 inches, as 
the water began to run down the face of the apparatus, and there was 
imperfect contraction of the jet. This head is roughly equal to one-half 
the diameter of the hole, but the effect occurs when the velocity of the jet 
is insufficient to prevent the adhesion of the water to the metal, and thus 
depends upon the absolute head, and not upon the ratio of head to 
diameter ; with a larger orifice the experiments could have been carried 
down to a relatively lower water surface. 
The most striking result, from a practical point of view, is that the 
curve of discharge is very nearly a straight line for values of the head 
between D and D/2. In fig. 3 the line through the points found experi- 
mentally has been drawn straight, and it represents, very closely, the 
results of the observations. The theoretical curve is straight between D and 
0’7D, but is curved between 07D and 05D, though the departure from 
the straight line is not great. Thus there would be a considerable simpli- 
fication in working out results if such a weir were adopted for gauging 
streams. The corresponding curve of discharge for a rectangular weir is of 
parabolic form, and only approximates to a straight line with relatively 
high heads of water. The circular weir has the disadvantage that it 
cannot be used for shallow streams, unless a deeper pool is excavated 
behind it. This is not always practicable, though it would be of great use 
in eliminating, to a considerable extent, the uncertainty in measuring the 
flow of the stream due to the “ velocity of approach ” of the water before it 
reaches the weir. 
The equation to the straight line, which represents approximately the 
discharge for any head between the centre and the top of the orifice, is 
easily obtained as 
Q = c(4*46 D 3/2 .H - T29 D 5/2 ), 
where Q is the actual discharge, in gallons per minute ; H is the head of 
water, measured in inches from the bottom of the orifice ; D is the diameter 
of the orifice, in inches ; and c is the coefficient of discharge, which may be 
taken as 0'625. For any particular size of aperture this simplifies to the form 
Q = aH - b, 
where a and b are numerical constants. The equation for the orifice used 
in the above experiments is 
Q = 11H - 8 gallons per minute (very nearly). 
