Page:Lowell Hydraulic Experiments, 4th edition.djvu/94

therefore,

$$a={{\log(l')-\log(l)} \over {\log(h)-\log(h'))}}.$$

We can thus, by means of two experiments, determine the power of the depth which will lead to identical quantities in the computed discharge of the two weirs.

123. It is assumed in the above equations, that the quantity discharged by a weir is directly proportioned to its length; this, in weirs having complete contraction, is, however, known not to be true, in consequence of the contraction which takes place at the ends of the weir. This contraction diminishes the discharge. When the weir is of considerable length in proportion to the depth of the water flowing over, this diminution is evidently a constant quantity, whatever may be the length, provided the depth is the same; we may, therefore, assume that the end contraction effectively diminishes the length of such weirs, by a quantity depending only upon the depth upon the weir. It is evident that the amount of this diminution must increase with the depth; we are unable, however, in the present state of the science, to discover the law of its variation; but experiment has proved that it is very nearly in direct proportion to the depth. As it is of great importance, in practical applications, to have the formula as simple as possible, it is assumed in this work that the quantity to be subtracted from the absolute length of a weir having complete contraction, to give its effective length, is directly proportional to the depth. It is also assumed that the quantity discharged by weirs of equal effective lengths, varies according to a constant power of the depth. There is no reason to think that either of these assumptions is perfectly correct; it will be seen, however, that they lead to results agreeing very closely with experiment.

124. The formula proposed for weirs of considerable length in proportion to the depth upon them, and having complete contraction, is

$Q=C(l-bnh)h^{a}$;^[1]

in which

${\textstyle Q=}$	the quantity discharged in cubic feet per second.
${\textstyle C=}$	a constant coefficient.
${\textstyle l=}$	the total length of the weir in feet.
${\textstyle b=}$	a constant coefficient.

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1. This formula was first suggested to the author by Mr. Boyden, in 1846.