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TABLE VI.-Breton's Sewage-Farm.

Summary of Crops gathered from March 25, 1875, to March 24, 1876, showing the quantity of each kind of Produce and Nitrogen contained therein.

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This acreage of Italian rye-grass includes not only the 17:33 acres of plots E, G, and H (marked in Table V.), but also the 12:12 acres of plot B, which were sown, according to the usual practice, for the following year's use, and from which only one very light cutting was taken.

1876.

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TABLE VII.-Breton's Sewage-Farm.

Statement of Land in crop and Land lying fallow on March 24, 1876.

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Improved Investigations on the Flow of Water through Orifices, with Objections to the modes of treatment commonly adopted. By Prof. JAMES THOMSON, LL.D., D.Sc.

[A communication ordered by the General Committee to be printed in extenso among

the Reports.]

THE methods usually put forward for treating of the flow of water out of vessels by orifices in thin plates, slightly varied though they may be in different cases, are ordinarily founded on assumptions largely alike in these different cases, and largely erroneous. The theoretical views so arrived at, and very generally promulgated, are in reality only utterly false theories based on suppositions of the flow of the water taking place in ways which are kinematically and dynamically impossible, and are at variance with observed facts of the flow, and even at variance with the facts as put forward by the advancers themselves of those theories. The admittedly erroneous results brought out through those fallacious "theories," and commonly miscalled "theoretical results," are afterwards considerably amended by the introduction into the formulas so obtained of constant or variable coefficients, or otherwise, so as to be brought into some tolerable agreement with experimental results. These means of practical amendment, however, being themselves not established on any scientific principles, can at best only conduce to the attainment of useful empirical formulas, but cannot, by their application to the originally false theoretical views, come to develop any true scientific theory. theory may, no doubt, be regarded as a good scientific theory, and as being good for practical purposes, which leaves out of account some minor features or conditions of the actual facts. In so far as it leaves any influential elements out of account, it is imperfect; but if the conditions which, for simplicity, or from want of complete knowledge of the subject, or for any other reason, are left out be of very slight influence on the practical results in question, the theory may be regarded as a very good one, though not quite perfect. In the case, however, of the hydraulic theories now referred to, the false principles involved in the reasonings relate to the main and important conditions of the flow, and not to any mere minor considerations, the imperfections or errors of which might be of but slight importance in the development of the main principles involved, and but little influential on the results sought to be attained.

A

I will now proceed to give some examples or sketches of the usual methods of treating the subject.

I will first take the case of water flowing from a state of rest through an orifice in a vertical plane face. This case is ordinarily treated by supposing the orifice to be divided into an infinite number of infinitely narrow horizontal bands of area, and supposing the velocity of the water in each band to be that due, through the action of gravity, to a fall from the still-water surfacelevel down to that band; then multiplying that velocity by the area of the band, and treating the product as being the volume flowing per unit of time across that horizontal band or element of the area; and integrating to find the sum of all these volumes of water for all the bands, and treating this sum as being the "theoretical" volume per unit of time flowing across the whole area of the orifice. This result is commonly called the "theoretical discharge" per unit of time; but, as it is known not to be the actual discharge, it is then multiplied by a numerical coefficient called by some "the

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