filaments, while the external filaments would fail to exert that necessary confining pressure. These external filaments could, with very little change in their own velocities, allow even of a great augmentation of the crosssectional area of the jet if the internal filaments, by abated velocity, were requiring to become considerably thicker than before, in virtue of the introduction of the obstruction. It is only the rapid change of direction of motion of the particles of water in the outer filaments in the neighbourhood of G, close to the obstruction, that enables them, by what may be called their centrifugal force, to maintain a greatly increased internal pressure very close to the obstruction, and so to allow of the water in the internal stream-filaments abating its velocity, and of those filaments themselves swelling in their transverse dimensions.

These considerations complete all that is necessary for the demonstration of Theorem II., and it may now be regarded as proved.

FORMULA FOR THE FLOW OF WATER IN THE V-NOTCH.

From the foregoing principle we can find intuitively the formula for the quantity of water which will flow through a V-notch in a vertical plane surface, as in fig. 11. We can see it at once by considering any stream-filament

in the flow in one notch, and the homologous stream-filament in the simila flow in another notch similarly formed, but having its vertex at a different depth below the still-water surface-level. Let the ratio of the depth of the vertex of the one notch below the still-water surface-level to the depth of the vertex of the other be as 1 to n, so that all homologous linear dimensions in the two flows will be likewise as 1 to n. Then, in passing from any cross section of one of the two homologous filaments to the homologous cross section of the other, we have the cross-sectional area o n2, and the velocity of flow

Vn; and the volume of water flowing per unit of time, being as the crosssectional area and the velocity conjointly, will vary as we pass from the one to the other of the pair of homologous filaments, so as to be a n2n. Then, as this holds for every pair of homologous stream-filaments throughout the two flows, if we put Q to denote the quantity, reckoned voluminally, flowing per unit of time in each of the two entire flows, we have

Qx nt.

Now, as well as considering two separate notches with different streams flowing in them at the same time, we may, when it suits our purpose, consider one single notch with streams of different depths flowing at different times; and if in various cases, either of the same V-notch or of different but

similar V-notches, we denote the height of the still-water surface-level above the level of the vertex of the notch by h, we have

where c is a constant coefficient, which cannot be determined by theory, but can be very satisfactorily determined by experiment for any desired ratio of horizontal width to vertical depth to be adopted for the form of the notch. Experiments determining the values of c for certain forms and arrangements of V-notches, suited for practical convenience and utility, have already been made by myself, and have been reported on to the British Association; and the Reports on them are printed in the British Association volume for Leeds Meeting, 1858, and in that for Manchester Meeting, 1861.

INVESTIGATION OF A FORMULA FOR THE FLOW OF WATER IN A RECTANGULAR NOTCH WITH LEVEL CREST IN A VERTICAL PLANE FACE.

It is to be premised that the long-known and generally used formulas for the flow of water in rectangular notches, brought out by the so-called "theories" which I have dissented from in the earlier part of the present paper, have been mainly of the form

Q=cgLh,

where Q denotes the volume per unit of time,

L denotes the horizontal length of the notch,

h the vertical height from the crest of the notch to the still-water surface-level, and

and where c has either been taken as a constant numerical coefficient for want of accurate experiments to determine its values for different values of L and h, or has been treated as a variable. Poncelet and Lesbros have taken this latter course, and have deduced by experiments extensive tables of its values for different depths of water in notches of the width on which they experimented—a width, namely, of 20 centimetres *. As, however, the coefficient for terrestrial gravity varies but little for different parts of the world, it has most frequently been left out of account, a single coefficient c' being used instead of cg; so that if, for instance, when the foot and second are used as units of length and time, we take 32.2 as a correct enough statement of the value of g for any part of the world, we have c'=32.2c.

A new formula, involving an important improvement in its form and adjusted so as to be in due accordance with numerous elaborate experiments, was developed within or about the time from 1846 to 1855, in America, by Mr. Boyden and Mr. Francis, both of Massachussetts. It is

Q=3·33 (L—nh)h3,

where Q is the quantity of water in cubic feet per second,

L is the length of the notch in feet,

h is the height from the level of the crest to the still-water surfacelevel in feet, and

n is the number of end contractions, and must be either 0, 1, or 2.

* Mémoires de l'Académie des Sciences: Sciences Mathématiques et Physiques, tome iii. 1829.

This formula was offered by Mr. James B. Francis, in his work entitled "Lowell Hydraulic Experiments," and published at Boston in 1855, not as one founded on any complete theoretical views, but as one depending on several assumptions probably not perfectly correct, and yet as one which, through numerous trials and by adjustments introduced tentatively in fitting it to experimental results, had been brought out so as to agree very closely with experiments.

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66

In § 120, at page 72 of his work, Mr. Francis says:-"No correct formula "for the discharge of water over weirs, founded upon natural laws, and in"cluding the secondary effects of these laws, being known, we must rely "entirely upon experiments, taking due care in the application of any formula "deduced from thence not to depart too far from the limits of the experiments "on which it is founded." And in §§ 123, 124, at page 74, in respect to the conception of the formula, he further gives the following very clear explanations: The contraction which takes place at the ends of a weir dimi"nishes the discharge. When the weir is of considerable length in proportion "to the depth of the water flowing over, this diminution is evidently a con"stant 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 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 [Mr. Francis's book] 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.

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

"in which Q=the quantity discharged in cubic feet per second;

"n=the number of end contractions. In a single weir having complete contraction, n always equals 2; and when the length of the weir is equal to the width of the canal leading to it, n = 0;

=

"h the depth of water flowing over the weir taken far enough upstream from the weir to be unaffected by the curvature in the surface caused by the discharge;

"a a constant power."

This formula, Mr. Francis states, was first suggested to him by Mr. Boyden in 1846.

The important novel feature in this formula consists in the subtraction which it makes, from the length L of the notch, of a length for each end

contraction directly proportional to the height of the still-water surface-level above the crest in order to find what may be treated in the formula as the effective length.

The formula in its general form here last noted expressed only in symbols, as also in its subsequently developed form here previously stated with numerical coefficients arrived at by tentative application of numerous experiments, is thus to be regarded as an ingeniously arranged and valuable empirical formula, but not as one founded on any trustworthy hydrokinetic theory. It is founded partly on the old ordinary false "theoretical" views, and partly on good conjectural assumptions, and is adjusted and approximately verified by elaborate experiments conducted on a scale unusually large, and with unusually good means for attainment of exact results. Mr. Francis, it is to be noticed, explains that, in the formula as finally brought out, the index for the power of the height of the water is taken as an exact fraction,, in preference to some unascertained fractional expression, different in no great degree from, merely for the attainment of facility in calculations in the practical applications of the formula, and not for any theoretic reason. Also it is to be noticed, in respect to the value which he assigned for the symbol b, that the symbol itself was first assumed as a constant rather than some unknown variable dependent on h, and was afterwards fixed at the particular value for the sake, in both cases, of attaining a convenient degree of simplicity which by trials was found to be attainable, consistently with good accordance between the representations afforded by the formula and the results shown by experiments. He supposed, however, that many other values of a and b (probably an unlimited number) might "be found that would accord somewhat nearer with the experiments"*.

Many years ago, after my having become acquainted with the empirical formula thus made out by Mr. Boyden and Mr. Francis, it occurred to me as desirable to attempt to investigate by hydrokinetic principles, without special experiments, a true formula for the flow of water in rectangular notches in vertical thin plates, or vertical plane faces, on the hypothesis of the water being a perfect or frictionless fluid, and by using in the formula symbols for constant coefficients, which, after the finding of the formula, might be determined by a small number of accurate experiments, and might further be tested as to their trustworthiness, or might be amended so as to become more exact, by a large number of varied experiments. It will be interesting to notice that the formula which had previously been arrived at in America by Mr. Boyden and Mr. Francis in the way already described is in perfect agreement with the formula which, by my own investigation, is brought out by strict scientific principles as a highly exact formula for water considered as a perfect fluid, and as being a very satisfactory representation of the truth for real water.

It is to be noticed at the outset that obviously a notch may be made so long relatively to the depth of its crest from the still-water surface-level, that, for any additional length, the increase of the flow will be proportional to the additional length. Let mh, in which m is a constant multiplier, be such a length as that, for additional length, the additional flow will be proportional to the addition made to the length. In fig. 12 let A B be the crest of the notch, and let CD be the level of the still-water surface of the pent-up water. Let A E and BF be each equal to mh, so that, over the part EF

* Lowell Hydraulic Experiments, § 156, p. 118; § 153, p. 116; and the passage quoted above from § 123, p. 74.

of the crest there will flow a quantity of water exactly proportional to the length of E F if the width of the notch be varied while the depth h of the water remains unchanged. Let the length EF be denoted by 1; then

l=L-mh.

Now, out of the entire flow, conceive the middle portion which flows over EF, and may be regarded as bounded laterally by two vertical planes perpendicular to the plane of the orifice, one passing through ER and the other through FS, to be taken away; and suppose the two remaining parts which flow over A E and B F, with the necessary lateral parts of the notch-plate, to be brought together as shown in fig. 13, so as to form one notch having

mh for its width and h for the height from crest to still-water level, and in which, therefore, the width of the notch shall bear a constant ratio to the height of the water, when the height varies, the width being always m times the height.

Then, by exactly the same mode of procedure as that already used for finding a formula to show how the quantity of water flowing in a V-notch varies with the depth of the vertex or with any other linear dimension of the flowing stream, we can readily see that if we put Q' to denote the volume of water flowing per unit of time in the case represented in fig. 13, we shall

have

where a is a constant coefficient.

Next to find an expression for the quantity (voluminally reckoned) flowing over the middle part E F of the crest, we may consider, first, of that middle part, a portion GK taken always of a length bearing a constant ratio to h; and for simplicity we may take it of length equal to h*. Now in this stream,

*Or, to meet the case in which there might not be, between E and F, a length so great ash, we might as well consider, in another notch having great width and having a height of flow equal to h, a portion of the flow not near either lateral extremity of the notch, and occupying a length of the crest equal to h.