from 14 cubic feet per minute to 39, the coefficient c remains almost absolutely constant; and thus the theoretic anticipation that the quantity should be proportional, or very nearly so, to the power of the depth is fully confirmed by experiment. The mean of these six values of c is 3064; but, being inclined to give rather more weight, in the determination of the coefficient as to its amount, to some of the experiments made this year than to those of last year, I adopt 305 as the coefficient, so that the formula for the rightangled notch without floor will be

Q=+305 H⭑.

My experiments on the right-angled notch with the level floor, fitted as already described, comprised the flow of water for depths of 2, 3, 4, 5, and 6 inches. They indicate no variation in the value of c for different depths of the water, but what may be attributed to the slight errors of observation. The mean value which they show for c is 308; and as this differs so little from that in the formula for the same notch without the floor, and as the difference is within the limits of the errors of observation, and because some consecutive experiments, made without and with the floor, indicated no change of the coefficient on the insertion of the floor, I would say that the experiments prove that, with the right-angled notch, the introduction of the floor produces scarcely any increase or diminution on the quantity flowing for any given depth, but do not show what the amount of any such small increase or diminution may be, and I would give the formula

Q=·305 H*

as sufficiently accurate for use in both cases. The experiments in both cases were made with care, and are without doubt of very satisfactory accuracy; but those for the notch without the floor are, I consider, slightly the more accurate of the two sets.

The experiments with the notch with edges sloping two horizontal to one vertical showed an altered feature in the flow of the issuing vein as compared with the flow of the vein issuing from the right-angled notch. The edges of the vein, on issuing from the notch with slopes two to one, had a great tendency to cling to the outside of the iron notch and weir-board, while the portions of the vein issuing at the deeper parts of the notch would shoot out and fall clear of the weir-board. Thus, the vein of water assumed the appearance of a transparent bell, as of glass, or rather of the half of a bell closed in on one side by the weir-board and enclosing air. Some of this air was usually carried away in bubbles by the stream at bottom, and the remainder continued shut up by the bell of water, and existing under slightly less than atmospheric pressure. The diminution of pressure of the enclosed air was manifested by the sides of the bell being drawn in towards one another, and sometimes even drawn together, so as to collapse with one another at their edges which clung to the outside of the weir-board. On the full atmospheric pressure being admitted, by the insertion of a knife into the bell of falling water, the collapsed sides would instantly spring out again. The vein of water did not always form itself into the bell; and when the bell was formed, the tendency to the withdrawal of air in bubbles was not constant, but was subject to various casual influences. Now it evidently could not be supposed that the formation of the bell and the diminution of the pressure of the confined air could occur as described without producing some irregular influences on the quantity flowing through the notch for any particular depth of flow, and this circumstance must detract more or less from the value of the wider notches as means for gauging water in compa rison with the right-angled notch with edges inclined at 45° with the hori

zon. I therefore made numerous experiments to determine what might be the amount of the ordinary or of the greatest effect due to the diminution of pressure of the air within the bell. I usually failed to meet with any perceptible alteration in the quantity flowing due to this cause, but sometimes the quantity seemed to be increased by some small fraction, such as one, or perhaps two, per cent. On the whole, then, I do not think that this circumstance need prevent the use, for many practical purposes, of notches of any desired width for a given depth.

My experiments give as the formula for the notch, with slopes of two horizontal to one vertical, and without the floor,

Q=0.636 H+,

and for the same notch, with the horizontal floor at the level of its vertex, Q=0·628 H.

In all the experiments from which these formulas are derived, the bell of falling water was kept open by the insertion of a knife or strip of iron, so as to admit the atmospheric pressure to the interior. The quantity flowing at various depths was not far from being proportional to the power of the depth, but it appeared that the coefficient in the formula increased slightly for very small depths, such as one or two inches. For instance, in the notch with slopes 2 to 1 without the floor, the coefficient for the depth of two inches came out experimentally 0-649, instead of 0.636, which appeared to be very correctly its amount for four inches' depth. It is possible that the deviation from proportionality to the power of the depth, which in this notch has appeared to be greater than in the right-angled notch, may be due partly to small errors in the experiments on this notch, and partly to the clinging of the falling vein of water to the outside of the notch, which would evidently produce a much greater proportionate effect on the very small flows than on great flows. The special purpose for which the wide notches have been proposed is to serve for the measurement of wide rivers or streams in cases in which it would be inconvenient or impracticable to dam them up deep enough to effect their flow through a right-angled notch. In such cases I would now further propose that, instead of a single wide notch, two, three, or more right-angled notches might be formed side by side in the same weir-board, with their vertices at the same level, as shown in the an

nexed figure. In cases in which this method may be selected, the persons using it, or making comparisons of gaugings obtained by it, will have the satisfaction of being concerned with only a single standard form of gaugenotch throughout the investigation in which they may be engaged.

By comparison of the formulas given above for the flows through the two notches experimented on, of which one is twice as wide for a given depth as the other, it will be seen that in the formula for the wider notch the coefficient 636 is rather more than double the coefficient 305 in the other. This indicates that as the width of a notch, considered as variable, increases from that of a right-angled notch upwards, the quantity of water flowing

increases somewhat more rapidly than the width of the notch for a given depth. Now, it is to be observed that the contraction of the stream issuing from an orifice open above in a vertical plate is of two distinct kinds at different parts round the surface of the vein. One of these kinds is the contraction at the places where the water shoots off from the edges of the plate. The curved surface of the fluid leaving the plate is necessarily tangential with the surface of the plate along which the water has been flowing, as an infinite force would be required to divert any moving particle suddenly out of its previous course*. The other kind of contraction in orifices open above consists in the sinking of the upper surface, which begins gradually within the pond or reservoir, and continues after the water has passed the orifice. These two contractions come into play in very different degrees, according as the notch (whether triangular, rectangular, or with curved edges) is made deep and narrow, or wide and shallow. From considerations of the kind here briefly touched upon, I would not be disposed to expect theoretically that the coefficient c for the formula for V-shaped notches should be at all truly proportional to the horizontal width of the orifice for a given depth; and the experimental results last referred to are in accordance with this supposition. I would, however, think that, from the experimental determination now arrived at, of the coefficient for a notch so wide as four times its depth, we might very safely, or without danger of falling into important error, pass on to notches wider in any degree, by simply increasing the coefficient in the same ratio as the width of the notch for a given depth is increased.

APPENDIX.-April 1862.

With reference to the comparison made, in the concluding sentences of the foregoing Report, between the quantities of water which, for any given depth of flow, are discharged by notches of different widths, and to the opinion there expressed, that we might, without danger of falling into important error, pass from the experimental determination of the coefficient for a notch so wide as four times its depth, to the employment of notches wider in any degree, by simply increasing the coefficient in the same ratio as the width of the notch for a given depth is increased, I now wish to add an investigation since made, which confirms that opinion, and extends the determination of the discharge, beyond the notches experimented on, to notches of any widths great in proportion to their depths. This investigation is founded on the formula for the flow of water in rectangular notches obtained from elaborate and careful experiments made on a very large scale by Mr. James B. Francis, in his capacity as engineer to the Water-power Corporations at Lowell, Massachusetts, and described in a work by him, entitled 'Lowell Hydraulic Experiments,' Boston, 1855 †. That formula, for either the case in which there are no end-contractions of the vein, or for that in which the length of the weir is great in proportion to the depth of the water over its crest, and the flow over a portion of its length not extending to either end is alone considered, is

where L1 = length of the weir over which the water flows, without end-contractions; or length of any part of the weir not extending to

the ends, in feet:

This condition appears not to have been generally noticed by experimenters and writers on hydrodynamics. Even MM. Poncelet and Lesbros, in their delineations of the forms of veins of water issuing from orifices in thin plates, after elaborate measurements of those forms, represent the surface of the fluid as making a sharp angle with the plate in leaving its edge. †The formula is to be found at page 133 of that work.

H-height of the surface-level of the impounded water, measured vertically from the crest of the weir, in feet:

and Q=discharge in cubic feet per second over the length L, of the weir. It is to be understood that, in cases to which this formula is applicable, the weir has a vertical face on the upstream side, terminating at top in a level crest; and the water, on leaving the crest, is discharged through the air, as if the weir were a vertical thin plate.

To apply this to the case of a very wide triangular notch:-Let A B C be

the crest of the notch, and AC the water level in the impounded pool. Let the slopes of the crest be each m horizontal to 1 vertical; or, what is the same, let the cotangent of the inclination of each side of the crest to the

horizon be=m. Let A E, a variable length, =x. Then E D= # Let

m

EG be an infinitely small element of the horizontal length or width from A to C. Then EG may be denoted by dx. Let q quantity in cubic feet per second flowing under the length x, that is, under A E in the figure. Then dq will be the quantity discharged per second between ED and G F. Then, by the Lowell formula just cited, we have

in which the constant quantity is to be put =0, because when x=0, q also =0. Hence we have

Let now H, height in feet from the vertex of the notch up to the level surface of the impounded water =BK in the figure. Then A K=m H2. Let also Q the discharge per second in the whole triangular notch twice the quantity discharged under A K. Then, by formula (2), we get

or

=

=

(3)

To bring the notation to correspond with that used in the foregoing Report, let Q=the quantity of water in cubic feet per minute, and H=the height of the water level above the vertex in inches.

This formula then gives, deduced from the Lowell formula, the flow in cubic feet per minute through a very wide notch in a vertical thin plate, when H is the height from the vertex of the notch up to the water level, in inches, and when the slopes of the notch are each m horizontal to 1 vertical.

As to the confidence which may be placed in this formula, I think it clear that, for the case in which the notch is so wide, or, what is the same, the slopes of its edges are so slight, that the water may flow over each infinitely small element of the length of its crest without being sensibly influenced in quantity by lateral contraction arising from the inclination of the edges, the formula may be relied on as having all the accuracy of the Lowell formula from which it has been derived; and I would suppose that when the notch is of such width as to have slopes of about four or five to one, or when it is of any greater width whatever, the deviation from accuracy in consequence of lateral contraction might safely be neglected as being practically unimportant or inappreciable.

This formula for wide notches bears very satisfactorily a comparison with the formulas obtained experimentally for narrower notches, as described in the foregoing Report. For slopes of one to one the formula was Q=305 H, and for slopes of two to one the formula was Q=636 H. To compare these with the one now deduced for any very slight slopes, we may express them thus:

For slopes of 1 to 1........

And for slopes of 2 to 1

While for any very slight slopes, or for any very
wide notches, the formula now deduced from
the Lowell one is ......

Q='305 m H*

Q='318 m H3

Q=·320 m H3.

The very slight increase from 318 to 320 here shown in passing from the experimental formula for notches with slopes of two to one, to notches wider in any degree-that slight change, too, being in the right direction, as is indicated by the increase from 305 to 318 in passing from slopes of one to one, to slopes of two to one-gives a verification of the concluding remarks in the foregoing Report; and this may serve to induce confidence in the application in practice of the formula now offered for wide notches.

Report on Field Experiments and Laboratory Researches on the Constituents of Manures essential to cultivated Crops. By Dr.AUGUSTUS VOELCKER, Royal Agricultural College, Cirencester.

IN a Report read at the Aberdeen meeting, and subsequently printed in the 'Transactions of the British Association,' will be found recorded a number of field experiments on turnips and on wheat. Similar experiments upon these two crops have since been continued from year to year, and a new series of field experiments has been undertaken on the growth of barley.

In connexion with these field trials I have made numerous laboratory experiments on the solubility of the various forms and conditions in which phosphate of lime is likely to be presented to growing plants, and have likewise studied to some extent the influence of ammoniacal salts and a few other saline combinations on the solubility of the various forms in which phosphate of lime occurs in recent and fossil bones, in apatite, and other phosphatic materials.

The present Report will comprehend two sections. In the first I shall give the results of my field experiments on turnips, wheat, and barley; in the second section reference will be made to the solubility of phosphatic materials in various saline liquids.