and LXXX.) will be equal, and they will balance each other. This may be shewn by two balls suspended freely and united by a cord, having the point of the cord which is directly above the centre of the table at distances from the balls reciprocally as their weight; or by two balls united by a wire, and resting in equilibrio on a forked support fixed in the centre of the tables, which will continue in equilibrio when the tables are turned. In like manner other cases may be confirmed by experiment. LEM. VII. If a body revolves freely in any orbit about an immoveable centre, and in an indefinitely small time describes any nascent arc; and the versed sine of the arc be drawn which may bisect the chord, and being produced may pass through the centre of force ; the centripetal force, in the middle of this arc, will be as the versed sine directly, and the square of the time inversely. Plate 4. Let two bodies revolve round their centre of force S, s; let QPM, qpm, be the nascent arcs described in any times, l,t; and let PB, pb, or QR, Aa, be the versed sines bisecting the chords, and when produced, passing through S the centre of force. Supposing the arcs MOP, NAP, to be described in the same time with different forces, C. c; by Prop. LXXII. Cor. 4. OR : Aa :: 0 :c. Hence, supposing the forces to be equal, QR is equal to Aa described in the same time; and (by Lem V., QR or Aa : qr : : Ap? : 9p”, that is, since the motion in the arcs is uniform, Aa : qr :: 1? : t? Therefore supposing both the times and forces different, and compounding these ratios, QR : qr ::Cx1' :cxo?; whence OR QR .qr. C:C Plate 4. COR. 1. If a body P, revolving about the centre S, describes a curve line APQ, and a right line ZPR touches that curve in any point P; and, from any other point of the curve, QR is drawn parallel to the distance SP, meeting the tangent in R; and QT is drawn perpendicular to the distance SP; the centripetal force will be reciprocally as the quantity, SP- X QT? V, if this be taken of that magnitude which it ultimately acquires, supposing the OR SP?xQT' spx " ; qr that is, C is to c reciprocally as OR gr or the centripetal forces are reciprocally as SP' x eu, COR. 2 x COR. 2. Hence, if any curvilinear figure APQ is given ; and therein a point S is also given, to which a centripetal force is perpetually directed; the law of centripetal force may be found, by which the body P, continually drawn back from a rectilinear course, will be retained in the perimeter of that figure, and will describe the same by a perpetual revolution. That is, we are to find the quantity SPPXQT, reciprocally proportional to OR this force. PROP. LXXXI. If equal bodies, revolving in ellipses, describe equal areas in equal times, their centripetal forces are to one another inversely as the squares of their distances from the foci of the ellipse towards which they tend. Let S be the focus : let a body P, tending towards S, describe a part of the ellipse PQ; Plate 4. join SP; draw QR to the tangent YZ, parallel to SP; join PC. and produce it to G. Fig. 4. Complete the parallelogram QxPR, produce Ox to v, Quis ordinately applied to GP; draw DK, a diameter parallel to YZ, and draw IH from the other focus H to SP parallel to YZ; join HP, and draw OT perpendicular to SP, as also PF to DK. EP is equal to the greater semiaxis AC. For, because CS is equal to CH, ES is equal to EI, (El. VI. 2.) whence EP is half the sum of PS, PI, that is of PS, PH, for (Simson's Conick Sect II. 11. Cor.) the angle IPR is equal to HPZ; whence (El I. 29.) the angle PIH is equal to PHI, and Pl is equal to PH ; and PS, PA, together, (Simson's Conick Sect. II 1.) are equal to the whole axis 2AC. EP therefore is equal to AC. Putting L for the principal latus rectum of the ellipse, L, (by definition) is equal to 2BC? L 2CB? (for AC : CB :: CB :, whence = L.) And LxOR : LxPv:: QR : Pv; AC AU LxQR : LxPv :: AC: PC, And, And, striking out the equal quantities, L XQR : QT? :: AC *L* PC : Gv CB%. Then substitute for AC x L its equal 2CB, and LXOR : QT” : : 2BC” X PC : Gu x BC or BC? X 2PC : Gu x BC* 2PC : Gu or But the points and P continually approaching without end, 2PC and Gu are equal : wherefore LxQR and Q1%, proportional to these are also equal. Multiply these equals SP? SP'xQ1 into and L x SP will become equal to OR OR Therefore (by Lem. VII Cor. I and 2.) the centripetal force is reciprocally as LxSP, that is, since I is a given quantity, as SP, or in a duplicate ratio of the distance SP. BOOK Def. I. FLUID is a body, the parts of which yield to any force impressed upon them, and easily move out of their places. A PROPOSITION I. The weight of fluids is as their quantities of matter. Since each particle of any fuid gravitates towards the earth, the greater is the number of particles, that is, the quantity of matter in any mass of fuid, the greater will be the weight of that mass. Exp. 1. The different pressures of different columns of fluid in the same vessel at differ. ent depths appear from the different quantities of duid discharged, at different depths, in the same time, from orifices of the same bore. If the air be exhausted from a tube in part filled with water, and the tube be closed a up, the solidity of the paruicles of water will be perceived from the sound produced by suddenly lifting up the tube. COR. 2. Cor. Fluids gravitate in Auids of the same kind. For they cannot lose the property of gravity which belongs to all bodies by such a change of situation. a Exp. Suspend a stopped phial from one arm of a balance, in a vessel of water, and balance it by weights from the opposite arm of the balance : upon unstopping the phial under water, a quantity of water will rush into it, by which the weight will be increased as much as the weight of water in the phial. PROP. II. When the surface of a fluid is level, the whole mass will be at rest. Plate 5: Let ABCD be a vessel containing water, the level surface of which is EF. Conceive the whole mass of Auid in the vessel to be divided into thin strata, or plates, RS, TV, XY, &c. lying horizontally one above another; and into small perpendicular columns GH. IK, LM, &c. contiguous to each other. In the stratum XY, and the columns IK, let m, n, be two adjacent drops. Neither of these drops can move towards the column in which the other is, without driving that other out of its place, because the fluid is supposed incompressible. But, with whatever force the particle m endeavours to displace the particle n, this force is counterbalanced by an equal and contrary effort on the part of n : because (Prop. I.} they are equally pressed by the equal columns above them : consequently the particles will be at rest. PROP. III. Any part of a fluid at rest presses, and is pressed, equally in all directions. For (Def 1) each particle is disposed to give way on the slightest difference of pressure : and, by supposition, each particle is pressed by the contiguous particles in such manner as to be kept at rest in its place: it is therefore pressed with an equal degree of force on all sides; and, consequently, (Book II. Prop. III.) it presses equally in all directions. COR. Hence the lateral pressure of a fluid is equal to the perpendicular pressure. This is one of the most extraordinary properties of fluids, and can be conceived to arise only from the extreme facility with which the component particles move among each other. Exp. 1. Into several tubes, bent near their lower ends in various angles, pour a sufficient quantity of mercury to fill the lower parts of their orifices; then dip them into a deep glass vessel filled with water, keeping the orifice of the longer legs above the surface: whilst the: tubes are descending the mercury will be gradually pressed upwards in the tubes, and the pressure will be equal at any given depth, whatever be the direction of the pressing column of fluid in the shorter leg of the tube. Oil may be used instead of mercury . 2. Dip an open end of a tube, having a very narrow bore, into a vessel of quicksilver; then, stopping the upper orifice with the finger, lift up the tube out of the vessel: a short column |