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.

* 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.

R

TABLE VII.-Breton's Sewage-Farm.

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

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 "theorics," 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. A 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.

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

coefficient of contraction," and by others the "coefficient of discharge," in order to find the actual discharge per unit of time.

Thus for the case of a rectangular orifice in a vertical plane face, as in fig. 1—where W L is the level of the still-water surface, and A B C D is the

orifice, with two edges A B and C D level, and E F is an infinitely narrow horizontal band extending across the orifice at a depth h below the stillwater surface-level, and having dh as its breadth vertically measured, while it has 7, the horizontal length of the orifice, as its length, and where, as shown in the figure, the depths of the top and bottom of the orifice below W L are denoted by h, and h, respectively-if q is put to denote the so-called "theoretical" volume per unit of time, and Q the actual volume per unit of time, it is commonly stated that

с

and then when c is put to denote the so-called "coefficient of contraction,” it is stated that the actual quantity flowing per unit of time is

It is then customary to deduce from this a formula for the case of water flowing in a rectangular notch open above, as in fig. 2, by taking h1 =0, and so deriving, for the open notch, the formula

Q for notch = cl√2g.h

2

(2)

These examples may suffice for indicating the nature of the method commonly advanced; and it may be understood that the same method with the necessary adaptations is usually given for finding the flow through circular orifices, triangular orifices, or orifices of any varied forms whatever.

Now this method is pervaded by false conceptions, and is thoroughly unscientific.

First. Throughout the horizontal extent of each infinitely narrow band of the area the motion of the water has not the same velocity, and has not

the same direction at different parts; and the assumption of the velocity being the same throughout, together with the assumption tacitly implied of the direction of the motion being the same throughout, vitiates the reasoning very importantly. It is thus to be noticed at the outset that the division of the orifice into bands, infinitely narrow in height, but extending horizontally across the entire orifice, cannot lead to a satisfactory process of reasoning, and that the elements of the area to be separately considered ought to be infinitely small both in length and in breadth.

Secondly. For any element of the area of the orifice infinitely small in length and breadth it is not the velocity of the water at it that ought to be multiplied by the area of the element to find the volume flowing per unit of time across that element, but it is only that velocity's component which is normal to the plane of the element that ought to be so multiplied.

Thirdly. Whether, for any element of the area of the orifice, we wish to treat of the absolute velocity of the water there, or to treat of the component of that velocity normal to the plane of the orifice, it is a great mistake to suppose that the velocity at the element is that due by gravity to a fall from the stillwater surface-level of the pent-up statical water down to the element. The water throughout the area of any closed orifice in a plane surface, with the exception of that flowing in the elements situated immediately along the boundary of the orifice, has more than atmospheric pressure; and hence it can be proved that it must have less velocity than that due to the fall from the still-water surface-level down to the element.

The foregoing may be illustrated by consideration of the very simple case of water flowing from a vessel through a rectangular orifice in a vertical plane face, two sides of the rectangle being level, and the other two vertical, and end contractions being prevented by the insertion of two parallel guide walls or plane faces, one at each end of the orifice, and both extending some distance into the vessel perpendicularly to the plane of the orifice, so that the jet of issuing water may be regarded as if it were a portion of the flow through an orifice infinitely long in its horizontal dimensions.

Thus if the jet shown in section in fig. 3a be of the kind here referred to, while W L is the still-water surface-level, the so-called "theoretical velocities" at the various depths in the orifice, which are dealt with as if they were in directions normal to the plane of the orifice, can be, and very commonly are, represented by the ordinates of a parabola as is shown in fig. 3b, where B D represents in magnitude and direction the "theoretical velocity" at the top of the orifice, CE the "theoretical velocity" at the bottom of the orifice, and FG that at the level of any point F in the orifice these ordinates being each made = √2gh, where h is the depth from the still-water surface down to the level of the point in the orifice to which the ordinate belongs. Then, under the same mode of thought, or same set of assumptions, the area of that parabola between the upper and lower ordinates (BD and C E) will represent what is commonly taken as the "theoretical discharge" per unit of time through a unit of horizontal length of the orifice. But this gives an excessively untrue representation of the actual conditions of the flow. Instead of the parabola, some other curve, very different, such as the inner curve sketched in the same diagram, fig. 3b, but whose exact form is unknown, would, by its ordinates, represent the velocity-components normal to the plane of the orifice for the various levels in the orifice, and its area would represent the real discharge in units of volume per unit of time through

*Theorem I., further on, will afford proof of this.