From this last by (4) and (3) we get

whence by (5)

nh-nk=h', or n(h, k)=h';

h=nh.

(7)

From this, if we put P and P' to denote total pressures on homologous small areas at U and U', it follows that

This holds good for any homologous places in any homologous guide-tubes, and so it holds for immediately adjacent places in any two contiguous guidetubes. Hence, in respect to any small element of the septum between two adjacent guided stream-filaments in Flow No. 1, considered comparatively with a homologous element of a septum in Flow No. 2, the homologous differential pressures in No. 1 and No. 2 will be as 1 to n3.

Thus the demonstration is now completed of all that is included in Proposition A; and we are ready to go forward to the demonstration of Theorem II., for which Proposition A was meant to be preparative. For this we have to observe that the conclusions arrived at in Proposition A hold good, no matter what may be the forms of the guide-tubes, provided that they be similar in both flows; and no matter what may be the distribution of pressures throughout a terminal interface crossing the assemblage of guidetubes in No. 1, provided that the homologous pressures throughout a homologous terminal interface in No. 2 be anyhow maintained severally n3 times those in No. 1. Hence, if in Flow No. 1 the guide-tubes be formed so that the water shall flow along exactly the same paths as if it were left unguided, and were left free to shoot away, past the interface at E, to a distance from the orifice great in proportion to the thickness of the issuing stream, without meeting any obstruction-and if the guide-tubes in No. 2 be similar to them--and if in No. 1 the system of pressures distributed throughout the terminal interface at E be made exactly the same as if the water were flowing freely for a great distance past that terminal interface—and if in No. 2 the system of homologous distributed pressures throughout a homologous terminal interface at E' be anyhow maintained severally n3 times those in No. 1,-it follows that the differential pressure on the two sides of any element of a septum in Flow No. 1 will be zero, as the guide-tubes have there no duty to perform. Then, on the homologous septum element in No. 2, the differential pressure, being n3 times that in No. 1, will be zero also. Hence in No. 2 the guide-tubes have no duty to perform, and the water flows in them exactly as if it were left unguided, but had still throughout its terminal interface the stated system of distributed pressures somehow applied.

Now, for completing the demonstration of Theorem II., nothing remains needed except to show that this stated system of distributed pressures requisite to be applied throughout the terminal interface at E' will very exactly be applied on that interface backwards by the water in front of it, which constitutes, for the time being, the continuation of the stream past that interface.

For proof of this, conceive any cross interface FF (fig. 9) further forward in No. 1 than EE is, and conceive a similarly situated cross interface FF in No. 2. By exactly the same mode of reasoning as before (making use of the like supposed introduction and subsequent removal of guide-tubes),

that reasoning being now applied to the two flows commencing at the initial interfaces BCD and B'C' D', and continued to the terminal interfaces F F and F' F', it results that if the jet in No. 1 be allowed to flow freely to and far past the interface FF, the jet in No. 2 terminating at F' F' can be left to flow unguided, with stream-lines similar to those in No. 1, and with the same relations of pressures and velocities at its various places to the pressures and velocities in No. 1 as have been already proved for the flow terminating at E' E', provided that homologous pressures n3 times those at FF be anyhow maintained at F' F'. Thus, then, we see that if adjusted or requisite pressure systems, such as have been already fully explained, be maintained at FF and F' F', the two streams, one extending backward from FF to EE, and the other from F' F' to E' E', will transfer backward just such pressures to successive places in retrograde order in their courses as that they will of themselves apply, at the interfaces E E and E' E', exactly the already specified requisite pressure systems. Thus we can depart

as far as we please from the orifices forward along the two streams to the places where, for purposes of reasoning, certain definite pressures are to be supposed to be applied in two homologous cross interfaces. Now it may be taken as evident that, by going far enough away from the orifice to the terminal cross interface, we can make, for any disturbances or departures from the specified pressure relations, the effects propagated backwards to the water in and near the orifices as small as we please; or that, even if we were to apply not exactly the strictly requisite pressure systems at those terminal places, still the effects of this departure from perfect exactitude would fade away rapidly in either stream as we transfer the place under consideration backwards against the current towards the orifice. In corroboration of this, observation on the flow of water spouting from an orifice may be appealed to as setting this matter beyond doubt, through its showing that any changes of pressure introduced in a jet of water at any place far 1876.

8

away from the orifice (as, for instance, by the insertion of a rigid obstruction) will transmit scarcely the slightest effect back to the region of the orifice; or, in other words, that in a free-flowing jet spouting through the air, the effects of obstructions fade away rapidly in the direction contrary to the current, so as to become imperceptible at a very moderate distance taken back from the obstacle in the direction against the flow-very moderate relatively to the thickness of the jet.

Even without this appeal to experimental observation, we might almost intuitively perceive, or might readily admit, that the introduction of more or less pressure than any stated amount in the stream, at a place where it has got well clear of the orifice, would be only very slightly influential on the flow as to pressures and as to velocities and directions of motion within the vessel and near the orifice and contracting vein. A reason for this is, that while an obstruction in a free jet will require a great change in mode of flow of the jet close in front of it, yet the jet approaching to that region need have its outer filaments turned aside only very slightly indeed to allow of all parts moving forward without any of their stream-lines, whether medial or at or near the surface, being subjected to almost any increase of pressure, and consequently without the velocities of any of them being almost at all retarded. This will readily be clearly understood by reference to fig. 10, where the water is shown as spouting against a stone without being

Fig. 10.

K

made to thicken its stream sensibly in consequence of the obstruction, except for a very short distance at G in front of the stone-that is to say, in the back-stream direction from the stone. If we were to suppose that the stone would have a tendency to produce, at such a place as K, any considerable increase of pressure in the internal or central stream-filaments of the jet, we would have to notice that the external stream-filaments next the atmosphere would fail to resist this augmented pressure; and, instead, they would, with only a very slight change in their own velocities or pressures, yield a little outwards, and so would not exert on the internal filaments the confining action that would be requisite for the maintaining of more than an extremely slight augmentation of pressure in those internal filaments. Then it is obvious that if the pressure is very little augmented, the velocity must be very little abated; and so, for this reason, the stream will not tend to thicken itself except very slightly, because any considerable increase of cross-sectional area of the stream would require an important abatement of velocity, which, as said before, would require a great increase of pressure in the internal

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 n2, and the velocity of flow

n; 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 nf.

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=cgLh1,

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

g the coefficient for gravity,

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,

1829.

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.