a unit of horizontal length of the orifice. Although the exact form of this true curve is unknown, yet we may observe that it must have its ordinates each less than the ordinate for the same level in the parabola. The truth of this may be perceived through considerations such as the following. First, it is to be noticed that for the very top and the very bottom of the orifice, instead of the ordinates B D and CE of the parabola, the ordinates of the true curve must be each zero; because, at each of these two places, the direction of the motion is necessarily tangential to the plane of the orifice, and so the velocity-component normal to the plane of the orifice * The assertion here made, to the effect that the directions of the stream-lines which form the external surface of the jet on its leaving the edge of the orifice must, at the edge, be tangential to the plane of the orifice when the orifice is in a plane face, or must in general be tangential to the marginal narrow band or terminal lip of the internal or waterconfining face of the plate or nozzle in which the orifice is formed, can be clearly and easily proved, although, strangely, the fact has been and is still very commonly overlooked. Even MM. Poncelet and Lesbros, in their delineations of the forms of veins of water issuing from orifices in thin plates, after elaborate observations and measurements of those forms, represent the surface of the issuing fluid as making a sharp angle with the plane wetted face in leaving the edge ("Expériences Hydrauliques sur les Lois de l'Écoulement de l'Eau," a Memoir read at the Academy of Sciences in November 1829, and published in the Mémoires, Sciences Mathématiques et Physiques, tome iii.). Other writers on Hydraulics put forward very commonly representations likewise erroneous. Weisbach, for instance, in his valuable works (Ingenieur und Maschinen-Mechanik, vol. i. § 313, fig. 427, date 1846; and Lehrbuch der theoretischen Mechanik, 5th ed. date 1875, edited by Hermann, § 433, fig. 772), has assumed (not casually, but with deliberate care, and after experimental measurements made by himself), as the best representation which, with available knowledge of the laws of contraction of jets of water, can be given for the form of the must be zero; and that component, not the velocity itself, is what the ordinate of the true curve must represent. On the hypothesis of perfect fluidity in the water (which, throughout the present discussions and investigations, is assumed as being a close enough representation of the truth to form a basis for very good theoretical views), the velocities at top and bottom of the orifice will be those due by gravity to falls from the still-water surfacelevel down to the top and bottom of the orifice respectively, because at these places the water issues really into contact with the atmosphere, and consequently attains atmospheric pressure. At all intervening points in the plane of the orifice it may readily be seen, or may with great confidence be admitted, that the pressure will be in excess of the atmospheric pressure; because, neglecting for simplicity the slight and, for the present purpose, unimportant modification of the courses of the stream-lines caused by the force of gravity acting directly on the particles composing the streamlines, as compared with the courses which the stream-lines would take if the action of gravity were removed, and the water were pressed through the orifice merely by pressure applied, as by a piston or otherwise, to the fluid in the vessel, we may say, truly enough for the present purpose, that an excess of pressure at the convex side of any stream-line is required in order that the water in the stream-line can be made to take its curved path. The mode of reasoning on this point suggested here may be obvious enough, although, for the sake of brevity, it is here not completely expressed. It follows that at all these intervening points in the plane of the orifice the absolute velocity of the water will be less than that due to a fall from the still-water surface down to the level of the point in the orifice; and besides, at all depths in the plane of the orifice except a single medial one, the direction of the flow will be oblique, not normal, to the plane of the orifice. Hence, further, through these two circumstances, jointly or separately as the case may be, it follows obviously that the ordinates of the true curve will everywhere be less than those of the parabola. Fig. 4 illustrates in like manner the false theoretical and the true actual conditions of the flow over a level upper edge of a vertical plane face, which may be exemplified by the case of a rectangular notch without end contractions, or of a portion of the flow not extending to either end in a very wide rectangular notch. In this case it is to be observed that the ordinates at and near the top of the issuing water in the vertical plane of the orifice must be only slightly less than those of the parabola-because, at the very top or outside of the stream, atmospheric pressure is maintained throughout the length of any stream-line, and so the velocity will be very exactly that due by gravity to the vertical depth of the flowing particle below the stillwater surface-level in the vessel; and because, also, the direction of the contracting vein of water issuing from a circular orifice in a thin plate, a solid of revolution specified clearly in such a way that the water surface in leaving the plane of the plate makes an angle of about 67° with that plane, and states to the effect that that water surface is just a continuation of the paths of the stream-lines within the vessel which he represents at the margin of the orifice as crossing the plane of the orifice with converging paths making the angle already mentioned of about 67° with that plane. They ought in reality to leave the lip tangentially to the plane, and then to make a very rapid turn in a short space (or to have a very small radius of curvature) on just leaving the lip of the orifice. The prevalence of erroneous representations and notions on this subject was adverted to, and an amendment was adduced, by myself in a Report to the British Association in 1861 on the Gauging of Water by V-Notches (Brit. Assoc. Rep. Manchester Meeting, 1861, part 1, p. 156). motion does not deviate much from perpendicularity to the plane of the orifice. Lower down in the plane of the orifice the direction of the water's motion will approach still more nearly to being perpendicular to that plane; but there the pressure will be considerably in excess of the atmospheric pressure, and so the velocity will be considerably less than that due by gravity to a fall through the vertical distance from the still-water surface-level down to the stream-line in the plane of the orifice. At places still further down in the orifice the flow comes to be obliquely upwards; and this obliquity is so great as to render the normal component very much less than the actual velocity, while the actual velocity itself is less than that due by gravity to the depth of the particle below the stillwater surface-level. At this region of the flow then, for both reasons, the ordinates of the true curve are less than those of the parabola. Lastly, at the very bottom of the orifice, or immediately over the top of the crest of the notch, the water issues into contact with the atmosphere, and so attains to atmospheric pressure, and must therefore have the velocity due by gravity to its depth below the still-water surface-level. Here, however, its direction of flow is necessarily tangential to the plane face of the vessel from which it is shooting away, and consequently is vertically upwards. Hence the normal component of its motion is zero, and so the ordinate of the true curve at that place is zero in length, instead of the normal component being greater at the bottom of the orifice than at any higher level, and instead of that component being properly represented by the ordinate there of the parabola. Like explanations to those already given might be offered for other forms of orifices (for circular or triangular orifices or V-notches, and for orifices in general which may be in vertical or horizontal or inclined plane faces, or in faces of other superficial forms than the plane), and it might be shown that in general the ordinary modes of treating the subject are very faulty. The examples already discussed may suffice to direct attention to the faulty character of the ordinarily advanced theories, and to give some suggestions of directions in which reforms are requisite. I will now proceed to offer some improved investigations which are appli cable to many of the most ordinary and most useful cases in practical hydraulics, in reference to the flow of water through orifices in thin plates, or from the wetted internal surface of vessels terminating abruptly in orifices. In devising and arranging these investigations I have aimed at putting them in such form as that they may be intelligible and completely demonstrative to students even in the early stages of their progress in dynamical studies. Definition.-The free level for any particle of water in a mass of statical or of flowing water is the level of the atmospheric end of a column, or of any bar straight or curved, of particles of statical water, having one end situated at the level of the particle, and having at that end the same pressure as the particle has, and having the other end consisting of a level surface of water freely exposed to the atmosphere, or else having otherwise atmospheric pressure there; or briefly we may say that the free level for any particle of water is the level of the atmospheric end of its column, or of an equivalent ideal pressure-column. pressure THEOREM I.-In the case of steady flow from approximate rest of water or any liquid considered as frictionless and incompressible, the velocity of any particle in the stream is equal to the velocity which a body would receive in falling freely from rest through a vertical space equal to the fall of free level which is incurred by the particle in the stream during its flow from rest to its existing position. Or, in briefer words sufficiently suggestive, it may be said that, in respect to water or any liquid flowing so as to admit of its being regarded as truly enough frictionless and incompressible, In steady flow, the velocity generated from rest is that due by gravity to the fall of free level. Or if be the fall of free-level sustained by any particle in passing from a statical region of the mass of water to a point in the region of flow, and if v be the velocity of the particle when at that point, then v=√2g5. In fig. 5, let W L be the still-water surface-level, and let B' BB" be a bounding interface separating the region of flow with important energy of motion from the region which may be regarded as statical, or as devoid of important energy of motion. Let U' U U" be another interface crossing the stream-lines at any place in the region of flow. Now taking as the unit of volume the cube of the unit of length, taking as the unit of area the square of the unit of length, taking the unit of density as unit of mass per unit of volume, so that the density of a body will be the number of units of mass per unit of volume, taking as the unit of force the force which acting on a unit of mass for a unit of time imparts to it a unit of velocity (that is to say, using the unit of force selected according to the system of Gauss, and which is often called the "absolute" or the "kinetic" unit of force *), and taking water-pressures as being reckoned from the atmospheric pressure as zero, let p density of the water; vvelocity at U; h-pressure-height at B, or the height of a column of statical water which would produce the pressure at B; h-pressure-height at U; Po=pressure in units of force per unit of area at B; P pressure in units of force per unit of area at U; f=fall from B to U, measured vertically; then and =fall of free level in the flow from the region of statical water to U; P-gph Let a small mass, m, of the water, whose volume (or content voluminally considered) is denoted by c, be introduced into the stream, its first place being at B just outside of the initial interface B' B B", and let it flow forward in the stream till it reaches a second place at U where it is just past the interface U'U U". In the stream filament BU E the space between the two interfaces at B and U is traversed alike by both front and rear of the small mass m; and therefore no excess of energy is given or taken by the mass in consequence of the pressure on its front and of that on its rear, for the passage of its front from the interface at B to that at U, and of its rear over the same space. as *The units of force derivable by the method of Gauss from the various units of length, mass, and time, in common use, though spoken of under general designations such "absolute units of force" or "kinetic units of force," have until lately been individually anonymous: and this deficiency, notwithstanding the important scientific and practical uses which these units were capable of serving, has been a great hindrance and discouragement to their general employment in dynamical investigations, and even to any satisfactory spread of knowledge of their meaning. Three years ago, the British-Association Committee on Dynamical and Electrical Units (Brit. Assoc. Report, 1873, part 1, p. 222), taking the centimetre, the gram, and the second as units of length, mass, and time, named the force so derived the Dyne. For the unit of force derived from the foot, the pound, and the second, the name Poundal has been introduced by myself; and it seems likely to come into use. At this Meeting of the British Association I have proposed the Crinal and the Funal as names for the two units of force derived respectively, one from the decimetre, the kilogram, and the second, and the other from the metre, the tonne, and the second (see Proceedings of Section A in the present volume). The familiarization of these important units to the minds of students of dynamics will, in a very important degree, aid the acquisition of clear and true views in hydrokinetics, as also in dynamics generally. |