But work given to it by pressure from behind, while it is passing the initial interface at B, is or that work is since pc=m. =P.c с =gmh¿, Again, during the emergence of the mass past the interface at U, it gives away to the water in front of it a quantity of work which, in like manner, is =p.c Also during the passage of the particle from its first place at B to its place at U it descends a vertical space = f; hence during that passage it receives from gravity a quantity of work=gmf. On the whole the mass receives an excess of work beyond what it gives, and that excess of work received is =gmh+gmf-gmh and as this is the work taken into store as kinetic energy, we have to put it which is the result that was to be proved in Theorem I. THEOREM II.-ON THE FLOW OF WATER THROUGH ORIFICES SIMILAR IN FORM AND SIMILARLY SITUATED RELATIVELY TO THE STILL-WATER SURFACE-LEVEL.—In the flowing of water, from the condition of approximate rest, through orifices similar in form and similarly situated relatively to the still-water surfacelevel*, the stream-lines in the different flows are similar in form: also the velocity of the water at homologous places is proportional to the square root of any homologous linear dimension in the different flows: and also (pressures being reckoned from the atmospheric pressure as zero) the pressure of the water on homologous small interfaces in the different flows is proportional to the cube of any homologous linear dimension; or, in other words, the fluid pressure (superatmospheric), per unit of area at homologous places, is proportional to any homologous linear dimension. Preparatively for the demonstration of this theorem, it is convenient to establish some dynamic principles, which, for present purposes, may be regarded as lemmas or preparatory propositions, and which will be grouped here together under the single heading of Proposition A. PROPOSITION A.-If there be two or more vessels containing water pent up in an approximately statical condition, and if they have similar orifices similarly situated relatively to the free level of the statical water—and if we imagine the *Or free level of the still water. water to be guided in each case to and onward past the orifice by an infinite number of infinitely small frictionless guide-tubes arranged side by side, like the cells of a honeycomb, and having their walls or septums of no thicknessand if, in the different vessels, these guide-tubes be, one set to another, similar in form, though they may be of quite different forms from the forms which the stream-lines would themselves assume if the flows were unguided—and if, at the homologous terminations of the guide-tubes, fluid pressures be anyhow maintained proportional, per homologous areas, to the cube of any homologous linear dimension, or, what is the same, if pressures be maintained proportional, per unit of area, to the homologous linear dimension,—then the velocity of the water at homologous places will be proportional to the square root of the homologous linear dimension, and the pressure of the water at homologous places on homologous areas will be proportional to the cube of the homologous linear dimension; and the water will press, at homologous places, on homologous areas of the septums, with a force on one side in excess of that on the other, which will be proportional to the cube of the homologous linear dimension. NOTE. For brevity in what follows, pressures at homologous places on homologous areas will be called homologous pressures, and pressures per unit of area will be called unital pressures; and any difference of the fluid pressures on the opposite sides of any small portion or element of a septum will be called a differential pressure. The demonstration of the proposition will be aided by first noticing the following relation in respect to two small solid masses in motion. If two similar small solid bodies of masses m and m', having their homologous linear dimensions as 1 to n, are guided to move along similar curves, having likewise their homologous linear dimensions as 1 to n (fig. 6), and if the velocities of the bodies at homologous points in their paths be as 1 to √, then First. Their gravities are as 1 to n3, evidently. Second. Their "centrifugal forces "+ applied by them in the plane of curvature and normally to the guide are also as 1 to n3. Let r and r' be put to denote the radii of curvature of the paths at homomv2 m'v'2 logous places. Then centrifugal forces are as : * The English form for the plural of septum, when septum is used as an English word, is here purposely preferred to the Latin septa. The name "centrifugal force" is here adopted in the sense in which it is commonly used. I fully agree with the opinion now sometimes strongly urged to the effect that this name is not a very happily chosen one; for two reasons:-first, because the name centrifugal would be better applied to a motion of flying from the centre, than to a force acting outwards along the radius; and secondly, because the body really receives no outward force, no force in the direction from the centre, but receives a centreward force which, being unbalanced, acts against the inertia of the body, and diverts the body from the straight line of its instantaneous motion. The centreward force actually received by the body, and which is the force acting on it normal to its path, may be called the deviative force received by the body. This is equal and opposite to the outward force called "centrifugal force," which is not received by the body, but is exerted outwards by it against whatever is compelling it to deviate from the straight line of its instantaneous motion. The name centrifugal force," however, although objected to, is in too general use throughout the world to allow of its immediate abandonment. ་་ This being understood, it readily becomes evident that if, instead of small solid masses sliding along guides, we have two small homologous masses of water m and m', fig. 7, flowing in similar slender guide-tubes, and if homo logous pressures be applied to the two masses in front and behind, which are as 1 to n3, and if at homologous situations in their two paths their velocities be as 1 to √, then, in respect to all the forces received by the two masses from without, other than those applied by the guide-tubes, and also in respect to the forces required to be received for counteracting their centrifugal forces, we see that all these constitute force systems similar in arrangement and of amounts as 1 to n3. It therefore follows that the forces which the masses must receive from their guide-tubes must be similarly arranged and of amounts, on homologous small areas, as 1 to n3. This being settled, we may now pass to the demonstration of Proposition A, at present in question. Suppose No. 1 and No. 2 in fig. 8 to represent two similar vessels with similarly guided flows, in all respects as described in the enunciation of this proposition. Let W L and W' L' be the still-water surface-levels, or the free levels of the still water in the two cases. Let BCD, B'C' D' be two similar bounding interfaces, each 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 BU E in No. 1 and B' U' E' in No. 2 be two homologous guide-tubes, and let them for the present be understood as terminating at two homologous cross interfaces E and E', which may conveniently be understood as being each situated at a moderate distance outside of the orifice for instance, at some such place as that which is usually spoken of as being the "vena contracta," or where the water has attained a pressure not differing much from that of the atmosphere, or it may in some cases even be that the atmospheric pressure is there attained; but the exact places at which to suppose the homologous terminations E and E' of the two guide-tubes as being taken are not at all essential to the demonstration. Let homologous linear dimensions in No. 1 and No. 2 be as 1 to n. Let the velocity at any variable point U in the guide-tube B U E be denoted by v. Let the pressure at U, expressed in units of pressure-height, be denoted by h; as shown by the vertical line U T in No. 1, where T is the top of the pressure-column for the point U. Let the pressure at B, the beginning of the tube, on the initial interface, outside of which the water may be regarded as statical, or as having no important energy of motion, be denoted by h; or, what comes to the same thing, let the depth from the still-water surface-level down to the beginning of the tube at B be denoted by ho, as is marked in the figure. It is thus to be noticed that the fall of free level incurred by a particle in flowing along the guide-tube from B to U is the vertical distance from the still-water surface-level, W L, down to T, the top of the pressure-column for the flowing water at U. This fall of free level may be denoted (in conformity with the notation in Theorem I.) by 4. Let the vertical descent from B to U be denoted by f; so that ƒ is the fall of a particle in passing from B to U. In case of an ascent in any guidetube, from its beginning to any point U in its course, we shall have the fall f negative. Let the abatement of pressure-height from B to U be denoted by k, or let ho-h-k. Thus in case of an increase of pressure-height in any guide-tube, from its beginning to any point U in its course, I will be negative. For No. 2, let the same letters of reference to the diagram, and the same notation, be used as for No. 1, with the modification for No. 2 merely of the attachment of an accent to each letter. Now as a part of the data on which the present investigation under Proposition A is founded, it is to be assumed that a unital pressure is somehow maintained at E', the end of the guide-tube in No. 2, n times that which is anyhow maintained at the corresponding point E in No. 1. Thus, if we denote these two pressures expressed as pressure-heights, at E and E' respectively, by h, and (he)', we have (h)'=nh.; and hence the fall of free level from beginning to end in No. 2 is n times the fall of free level from beginning to end in No. 1. Hence putting v. and (v)' to denote the velocities at E and E' respectively, we have (by Theorem I., which proves that the velocities must be proportional to the square roots of the falls of free level) or v: (v.)':: VI: √n, (1) Again, from similarity of forms, we have in respect to areas of crosssections of the two guide-tubes : This applies to any or all homologous points in the two regions of flow. Now by referring to the figure or otherwise, it will readily be seen that , or the fall of free level from B to U, is h2+f-h, while k=h-h; and that therefore=f+k. Hence, by Theorem I., we have Also, since the pressure at any point in a stream-line, or guide-tube, is its initial pressure minus the relief of pressure, we have |