1908-9.] Discharge of Water from Circular Weirs. 
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XVI. — On the Discharge of Water from Circular Weirs and Orifices. 
By G. H. Gulliver, B.Sc., A.M.I.Mech.E., Lecturer in Engineering 
in the University of Edinburgh. 
[(MS. received October 9, 1908. Read November 2, 1908.) 
Many investigators have measured the flow of water through orifices, and 
over weirs of various shapes and sizes. The discharge through a vertical 
circular orifice, situated at distances below the water surface much greater, 
relatively, than the diameter of the aperture, has been made the subject 
of study on several occasions, but the case of a circular weir, or notch — 
that is, a circular hole discharging across only a portion of its area — has 
been neglected hitherto. The author has had occasion recently to investi- 
gate the discharge of such a circular weir, more especially with regard to 
its suitability for measuring the flow of streams, a purpose for which weirs 
of rectangular shape are used almost exclusively at present. The problem 
forms an interesting example of the utility of graphical methods in cases 
for which solutions would be unobtainable by purely mathematical means. 
1. Circular Weir, or orifice partly drowned. 
Fig. 1 represents an orifice of radius K. Let AB be the water surface, 
which is situated at a height H from the lowest point of the orifice. Let 
CD be ail elementary strip of vertical thickness dh, and at a mean depth 
h below the water surface. 
The area of the strip, CD, is 2CF.cZ/i. 
But 
(CF) 2 4- (OF) 2 = (OC) 2 = R 2 , 
and 
OF = OK + KF = (R - H) + 7*; 
therefore 
(CF) 2 = R 2 - (R - H + ft) 2 
= 2R(H - ft) - (H - ft,) 2 
= (H - ft){D - (H - ft)}, 
where D, the diameter of the orifice, is written instead of 2R. 
The area of the elementary strip, CD, is 
2 iS /(H-ft){D-(H -h)}.dh. 
The discharge of water across this area, neglecting losses due to friction 
and contraction of the jet, is 
2 J(R - ft,){D - (H - ft,)} .dli. J2gh cubic feet per second. 
