PROPOSITION XXXVII. THEOREM XXIX. If a cylinder move uninformly forward in a compressed, infinite, arid non-elastic finid, in the direction of its length, the resistance arising from its the magnitude of its transverse whole motion moves four times may section is to the force by be destroyed or generated, in the its length, as the density of the which time that medium sity of the cylinder, nearly. For let the vessel touch the surface of stagnant water with bottom CD, and let the water run out of this vessel into the stagnant ABDC ter let EFTS it den to the its wa and through the cylindric canal perpendicular co the horizon the little circle PQ, be placed parallel to the horizon any where in the ;OF NATURAL PHILOSOPHY. SEC. VII.] middle of the canal AK that ratio, EF be to CK above the it K so I JL jg f -^ orifice of the canal PQ bears to the cir manifest (by Case is K, "" little circle Then to in the duplicate of the which the excess of the AB. cle may CA and produce ; 339 Case 5, e 1, Prop. XXXVI) that the velocity 6, of the water passing through the annular space between the little circle and the sides of the ves and Cor. be the very same which the water would acquire by falling, and in its fall describing the or IG. altitude sel will KG And (by Cor. 10, Prop. XXXVI) if the breadth of the vessel be infinite, HG HI may vanish, arid the altitudes IG, become equal the force of the water that flows down and presses upon the circle will be so that the lineola ; weight of a cylinder whose base is that little circle, and the altitude 2 2 iIG, as EF to EF |PQ 2 very nearly. For the force of the water flowing downward uniformly through the whole canal will be the same to the , upon the I ,et little circle now PQ. in whatsoever part of the canal it be placed. the orifices of the canal ST EF, be closed, and let the littk compressed on every side, and by its ascent let it oblige the water that lies above it to descend through the annular space between the little circle and the sides of the canal. Then will the velocity circle ascend in the fluid of the ascending little circle be to the velocity of the descending water as and PQ, is to the circle PQ; and the ve the difference of the circles EF locity of the ascending little circle will be to the sum of the velocities, that is, to the relative velocity of the descending water with which it passes by EF PQ the little circle in its ascent, as the difference of the circles to and 2 2 the circle EF, or as EF* to Let that relative velocity be equal to the velocity with v/hich it was shewn above that the water would EF PQ . pass through the annular space, if the circle were to remain unmoved, that is, to the velocity which the water would acquire by falling, and in its fall and the force of the water upon the ascending- describing the altitude IG ; as before (by Cor. 5, of the Laws of Motion) that the resistance of the ascending little circle will be to the weight of a circle will be the is, same ; cylinder of water whose base EF 2 to EF 2 iPQ 2 , is that little circle, and be to the velocity which scribing the altitude [G, as EF PQ 2 2 to EF Let the breadth of the canal be increased in between EF 2 PQ 2 its altitude iIG, as But the velocity of the little circle will nearly. the water acquires by falling, and in its fall de and EF 2 , and between 2 . wfinitum ; and the EF 2 and EF 2 ratios iPQ 2 . become at last ratios of equality. And therefore the velocity of the little circle w ill now be the same which the water would acquire in falling, and in its fall describing the altitude IG= and the resistance will become will rTHE MATHEMATICAL PRINCIPJ ES 340 [BOOK IT. equal to the weight of a cylinder whose base is that little circle, and its altitude half the altitude IG, from which the cylinder must fall to acquire the velocity of the ascending circle and with this velocity the cylinder in ; the time of its fall will describe four times its length. But the resistance of the cylinder moving forward with this velocity in the direction of its is the same with the resistance of the little circle (by Lem. IV), and therefore nearly equal to the force by which its motion may be generated while it describes four times its length. length is If the length of the cylinder be and the time in which or diminished in the ; its motion, same augmented & and therefore the force by which the mo ratio, tion so increased or diminished, tinue the same augmented or diminished, describes four times its length, O t will be it may because the time is be destroyed or generated, will con increased or diminished in the same proportion and therefore that force remains still equal to the resistance of the cylinder, because (by Lem. IV) that resistance will also remain the ; same. If the density of the cylinder be augmented or diminished, its motion, motion may be generated or destroyed in the same time, will be augmented or diminished in the same ratio. Therefore the resistance of any cylinder whatsoever will be to the force by which its whole motion may be generated or destroyed, in the time during which it and the force by which moves four times its length, as the cylinder- nearly. A fluid its the density of the medium to the density of Q..E.D. must be compressed to become continued; it must be continued and non-elastic, that all the pressure arising from its compression may be propagated in an instant and so, acting equally upon all parts of the body ; moved, may produce no change of the resistance. The pressure arising from the motion of the body is spent in generating a motion in the parts of the fluid, and this creates the resistance. But the pressure arising from the compression of the fluid, be it ever so forcible, if it be propagated in an no motion in the a of continued instant, generates fluid, produces no parts at all of motion therein and therefore neither change augments nor les ; sens the resistance. This is certain, that the action of the fluid arising from the compression cannot be stronger on the hinder parts of the body moved than on its fore parts, and therefore cannot lessen the resistance de scribed in this proposition. And if its propagation be infinitely swifter than the motion of the body pressed, it will not be stronger on the fore But that action will be infinitely parts than on the hinder parts. swifter, and propagated in an instant, if the fluid be continued and non- elastic. COR. 1. The resistances, made to cylinders going uniformly forward in the direction of their lengths through continued infinite mediums are in aOF NATURAL PHILOSOPHY- SEC. VII.] ratio 341 compounded of the duplicate ratio of the velocities and the duplicate and the ratio of the density of the mediums. ratio of the diameters, breadth of the canal be not infinitely increased but the in the direction of its length through an included forward cylinder go axis all the while coinciding with the axis of the its quiescent medium, will be to the force by which its whole motion, in the resistance its canal, K …………. I… …….. L time in which it describes four times its COR. 2. If the length, be generated or destroyed, in a ratio com 2 2 i to pounded of the ratio of may EF A E Hi EF 2 to EF 2 PQ, 2 once, and the ratio of EF twice, and the ratio of the density of the medium to the density of the cylinder. The same thing supposed, and that a of the length of the quadruple length the cylinder in a ratio compounded of the ratio 2 EF 2 – iPQ 2 to once, and the ratio of COR. 3. L EF 2 is to PQ, 2 EF EF 2 twice; the resistance of the cylinder will be to the force by which its whole motion, in the time during which it describes the length L, may be destroyed or generated, as the density of the medium to the density of the cylinder. to SCHOLIUM. In this proposition we have investigated that resistance alone which from the magnitude of the transverse section of the cylinder, neg lecting that part of the same which may arise from the obliquity of the arises motions. tions with as, in Case 1, of Prop. XXXVL, the obliquity of the mo which the parts of the water in the vessel converged on every For side to the hole EF hindered the efflux of the water through the hole, so, in this Proposition, the obliquity of the motions, with which the parts of the water, pressed by the antecedent extremity of the cylinder, yield to the pressure, and diverge on all sides, retards their passage through the places that lie round that antecedent extremity, toward the hinder parts of the cylinder, and causes the fluid to be moved to a greater distance; which in creases the resistance, and that in the same ratio almost in which it dimin ished the efflux of the water out of the vessel, that is, in the duplicate ratio And as, in Case 1, of that Proposition, we made the of 25 to 21, nearly. perpendicularly and in the parts of the water pass through the hole greatest plenty, by supposing all the water in the vessel lying round the EF cataract to be frozen, and that part of the water whose motion was oblique, and useless to remain without motion, so in this Proposition, that the obliquity of the motions may be taken away, and the parts of the water give the freest passage to the cylinder, by yielding to it witli the most direct and quick motion possible, so that only so much resistance may re- mayTHE MATHEMATICAL PRINCIPLES 542 main as arises from the magnitude of the transverse [BoOK section, and which II. is incapable of diminution, unless by diminishing the diameter of the cylinder we must conceive those parts of the fluid whose motions are oblique and useless, and produce resistance, to be at rest among themselves at both ex ; and there tremities of the cylinder, to cohere, and be joined ABCD be a rectangle, and let AE and BE be two parabolic arcs, to the cylinder. Let described with the axis AB, and with a latus rectum that is to i 1 g the j^ .+++-"" space HG, which must be described by the cylinder in falling, in order to acquire the velocity with which it moves, as HG to ^AB. Let CF and DF be two other parabolic arcs described with the axis CD, and a latus rectum quadruple of the former; and by the convolution of the figure ABDC ABE EF let there be generated a solid, whose middle part the cylinder we are here speaking of, and whose extreme parts and contain the parts of the fluid at rest among themselves, and concreted about the axis is CDF into two hard bodies, adhering to the cylinder at each end like a head and Then if this solid move in the direction of the length of tail. EACFDB FE the parts beyond E, the resistance will be the same which we have here determined in this Proposition, nearly that is, it will have the same ratio to the force with which the whole motion of the cyl its axis toward ; be destroyed or generated, in the time that it is describing the with that motion uniformly continued, as the density of the length And (by Cor. 7, Prop. fluid has to the density of the cylinder, nearly. inder may 4AC XXXVI) the resistance must be to this force in the ratio of 2 to 3, at the least. Lemma V. If a cylinder, a sphere, and a spheroid, of equal breadths be placed suc cessively in the middle of a cylindric canal, so that their axes may coincide with the axis of the canal, these bodies will equally hinder t^e passage of the water through the canal. For the spaces lying between the sides of the canal, sphere, and spheroid, through which the water and the cylinder, and the passes, are equal ; water will pass equally through equal spaces. This is true, upon the supposition that all the water above the cylinder, or whose sphere, spheroid, fluidity is not necessary to make the passage of the water the quickest possible, is congealed, as was explained above in Cer 7, Prop. XXXVI.Lemma The same 343 OF NATURAL PHILOSOPHY SEC. VII.] VI. are equally supposition remaining, the fore- mentioned bodies the canal. water the OIL acted Jlowin g through by This appears by Lein. V and the third Law. For bodies act upon each other mutually and equally. Lemma tht water and the VIL If the water be at rest in the canal, and these bodies move with equil ve locity and the contrary way be equal among themselves. This appears from the same among themselves. last through the canal, Lemma, their resistances will for the relative motions remain the SCHOLIUM. The case is the same of all convex and round bodies, whose axes coincide with the axis of the canal. Some difference may arise from a greater or but in these less friction; Lemmata we suppose the bodies to be perfectly and that smooth, and the medium to be void of all tenacity and friction those parts of the fluid which by their oblique and superfluous motions may ; disturb, hinder, rest amorg and retard the flux of the water through the canal, are at being fixed like water by frost, and adhering to themselves ; the fore and hinder parts of the bodies in the manner explained in the Scholium of the last Proposition for in what follows we consider the very : round bodies described with the greatest given trans verse sections can possibly meet with. least resistance that Bodies swimming upon fluids, when they move straight forward, cause the fluid to ascend at their fore parts and subside at their hinder parts, and thence they meet with a especially if they are of an obtuse figure ; little more resistance than bodies moving if they were acu*-e at the head and in elastic fluids, if they are obtuse behind dense the fluid a little more at their fore parts, and tail. And before, con and relax the same at theii hinder parts and therefore meet also with a little more resistance than ii But in these Lemmas and Proposi they were acute at the head and tail. ; we are not treating of elastic but non-elastic fluids; not of bodies And floating on the surface of the fluid, but deeply immersed therein. when the resistance of bodies in non-elastic fluids is once known, we may tions then augment this resistance a little in elastic fluids, as our air; and in the surfaces of stagnating fluids, as lakes and seas.