centre of motion A, the vibrations of pendulums would never cease. But from those obstructions, though small, it happens, that the velocity of the balk-in the point B is a little diminished in every vibration; and, consequently, it does not return precisely to the same points C or D, but the arcs described continually become shorter and shorter, till at length they grow insensible; unless the motion be assisted by a mechanical contrivance, as in clocks, called a maintaining power. ABA equal to the circumference of the circle, while the point A in the circumference traces out a curve line ACAGA: then this curve is called a cycloid; and some of its properties are contained in the following lemma: LEMMA. 47. If the generating or revolving circle be placed in the middle of the cycloid, its diameter coinciding with the axis AB, and from any point there be drawn the tangent CF, the ordinate CDE perpendicular to the axis, and the chord of the circle AD; then the chief properties are these: The right line CD - The semi-cycloid ACA = the circular arc AD; double the chord AD; double the diameter AB, and The tangent CF is parallel to the chord AD. PROP. XXVII. 48. When a pendulum vibrates in a cycloid, the time of one vibration, is to the time in which a body falls through half the length of the pendulum, as the circumference of a circle is to its diameter. Let ABa be the cycloid; DB its axis, or the diameter of the generating semicircle DEB; CB= 2DB the length of the pendulum, or radius of curvature at B. Let the ball descend from F, and, in vibrating describe the arc FBf. Divide FB into innumerable small parts, one of which is Gg; draw FEL, GM, On LB degm, perpendicular to DB. scribe the semicircle LMB, whose centre is 0; draw MP parallel to DB; also draw the chords BE, BH, EH, and the radius OM. Now, the triangles BEH, BHK, are similar; therefore, BK: BH :: BH : BE, or BHBK. BE, or BH = √ BK. BE. Also, the similar triangle MmP, MON, give Mp: Mm :: MN: MO. And, by the nature of the cycloid, Hħ is equal and parallel to Gg. If another body descend down the chord EB, it will have the same velocity as the ball in the cycloid has at the same height. So that Kk and Gg are passed over with the same velocity, and consequently the time in passing them will be as their lengths Gg, Kk, or as Hh to Kk, or BH to BK by similar triangles, or ✓BK. BE to BK, or √BE to √BK, or as BL to BN by similar tri angles. That is, the time in Gg: time in Kk :: √✓✓/BL: √BN. Again, the time of describing any space with a uniform motion, is directly as the space, and reciprocally as the velocity; also, the velocity in K or Kk, is to the velocity at B, as √EK to √EB, or as √LN: √LB; and the uniform velocity for EB is equal to half that at the point B, therefore the time in Kk : time Consequently, the sum of all the times in all the Gg's is to the time in EB, or the time in DB, which is the same thing, as the sum of all the Mm's is to LB; That is, the time in Fg : time in DB :: And the time in Or the time in Im : LB. That is, the time of one whole vibration, is to the time of falling through half CB, as the circumference of any circle, is to its diameter. Corol. Hence all the vibrations of a pendulum in a cycloid, whether great or small, are performed in the same time; which time is to the time of falling through the axis, or half the length of the pendulum, as 3.1416 to 1, the ratio of the circumference to its diameter; and hence that time is easily found thus. Put p = 3·1416, and 7 the length of the pendulum, also g the space fallen through by a heavy body in 1" of time: Theng: : : 1′′ : √/- the time of falling through, 49. And if the pendulum vibrate in the small arc of a circle; because that small arc nearly coincides with the small cycloidal arc at the vertex B; therefore the time of vibration in the small arc of a circle, is nearly equal to the time of vibration in the cycloidal arc; and consequently the time of vibration in a small circular arc is equal to p√~~~ where is the radius of the circle. 50. So that, if one of these, g or l, be found by experiment, this theorem will give the other. Thus, ifg or the space fallen through by a heavy body in I" of time, be found, then this theorem will give the length of the seconds pendulum. Or, if the length of the seconds pendulum be observed by experiment, which is the easier way; this theorem will give the descent of gravity in l". Now, in the latitude of London, the length of a pendulum which vibrates seconds, has been found to be 39 inches; and this being written for 7 in the theorem, 391 it gives p√ = 1′′; and hence is found 9= {p3l= }p3 × 39} = 193′07 9 inches 16 feet, for the descent of gravity in 1"; which it has also been found to be very exactly, by many accurate experiments. SCHOLIUM, * 51. Hence is found the length of a pendulum that shall make any number of vibrations in a given time. Or, the number of vibrations that shall be made by a pendulum of a given length. Thus, suppose it were required to find the length of a half seconds pendulum, or a quarter seconds pendulum; that is, a pendulum to vibrate twice in a second, or 4 times in a second. Then, since the time of vibration is as the square root of the length, Therefore 1 :: √39 : √l, 2 Or 1:39: seconds pendulum. 391 4 9 inches nearly, the length of the half And 1: 39: 2 inches, the length of the quarter seconds pendulum Again, if it were required to find how many vibrations a pendulum of 80 inches long will make in a minute, Here 80: 39:: 60′′ or l′ : 60√ almost 42 vibrations in a minute. 39 80 = 7/31·3 = 41.95987, or 52. In these propositions, the thread is supposed to be very fine, or of no sensible weight, and the ball very small, or all the matter united in one point; also, the length of the pendulum, is the distance from the point of suspension, or centre of motion, to this point, or centre of the small ball. But if the ball be large, or the string very thick, or the vibrating body be of any other figure; then the length of the pendulum is different, and is measured, from the centre of motion, not to the centre of magnitude of the body, but to such a point, as that if all the matter of the pendulum were collected into it, it would then vibrate in the same time as the compound pendulum; and this point is called the Centre of Oscillation, which will be treated of in what follows. The pendulum may be applied to three several important purposes. (1.) To measure portions of time, or to subdivide the units we derive from astronomical phenomena, into smaller and equal portions. (2.) To determine the measure of the force of gravity, at different places, and under different circumstances; and thus to enable us to infer the variation in the apparent intensity that is due to the centrifugal force; and the variation in the actual intensity at the surface, that is due to the figure of the earth. Hence the figure of the earth may be determined. (3.) The standard unit from which all lineal measures are taken, is the length of a pendulum vibrating seconds of mean time in the latitude of London, in a vacuum at the level of the sea, Fah. thermometer being at 62°, and the barometer at 30 inches. OF THE CENTRES OF PERCUSSION, OSCILLATION, AND GYRATION. 53. THE Centre of Percussion of a body, or a system of bodies, revolving about a point, or axis, is that point, which striking an immoveable object, the whole mass shall not incline to either side, but rest as it were in equilibrio, without acting on the centre of suspension. 54. The Centre of Oscillation is that point, in a vibrating body, in which if any body be placed, or if the whole mass be collected, it will perform its vibrations in the same time, and with the same angular velocity, as the whole body, about the same point or axis of suspension. 55. The Centre of Gyration, is that point, in which, if the whole mass be collected, the same angular velocity will be generated in the same time, by a given force acting at any place, as in the body or system itself. 56. The angular motion of a body, or system of bodies, is the motion of a line connecting any point and the centre or axis of motion; and is the same in all parts of the same revolving body. And in different, unconnected bodies, each revolving about a centre, the angular velocity is as the absolute velocity directly, and the distance from the centre inversely; so that, if their absolute velocities be as their radii or distances, the angular velocities will be equal. PROP. XXVIII. 57. To find the centre of percussion of a body, or system of bodies. LET the body revolve about an axis passing through any point S in the line SGO, passing through the centres of gravity and percussion, & and O. Let MN be the section of the body, or the plane in which the axis SGO moves. And conceive all the particles of the body to be reduced to this plane, by perpendiculars let fall from them to the plane; a supposition which will not affect the centres G, O, nor the angular motion of the body. S M B N Let A be the place of one of the particles, so reduced; join SA, and draw AP perpendicular to AS, and Aa perpendicular to SGO: then AP will be the direction of A's motion, as it revolves about S; and the whole mass being stopped at O, the body A will urge the point P forward, with a force proportional to its quantity of matter and velocity; or to its matter and distance from the point of suspension S; that is, as A. SA; and the efficacy of this force in a direction perpendicular to SO, at the point P, is as A. Sa, by similar triangles; also, the effect of this force on the lever, to turn it about O, being as the length of the lever, is as A. Sa. PO = A. Sa. SO- SP A. Sa. SO — A. Sa. SP A. Sa. SO _A. SA'. In like manner, the forces of B and C, to turn the system about O, are as, But, since the forces on the contrary sides of O destroy one another, by the definition of this force, the sum of the positive parts of these quantities, must be equal to the sum of the negative parts, that is, A. Sa. SO + B. Sb. SO + C. Sc. SO, &c. = A. SAB. SB + C. SC2, &c. ; and hence SO = the centre of percussion below the axis of motion. which is the distance of And here it must be observed that, if any of the points a, b, &c. fall on the contrary side of S, the corresponding product A. Sa, or B. Sb., &c. must be made negative. Corol. 1. Since, by cor. 3, pr. 15, A+ B+ C, &c. or the body b × the distance of the centre of gravity, SG, is = A. Sa + B. Sb + C. Sc, &c. which is the denominator of the value of SO; therefore the distance of the cenA. SA2 + B. SB2 + C. SC2 &c. SG X body b tre of percussion is SO = and, by cor. 5, pr. 13, the sum of the last terms is nothing, namely, 2SG. Ga + 2SG . Gb + 2SG, Gc, &c. = 0; therefore the sum of the others, or A. SA2 + B . SB2, &c. = A + B, &c. . SG2 + A. GA3 + B. GB3 + C . GC3, &c. or = b. SG2 + A GAB. GB3 + C . GC3, &c. which being substituted in the numerator of the foregoing value of SO, gives, b. SG2 + A. GA2 + B . GB2 + &c. SO = or SO SG + b. SG A. GA2+ B. GB2 + C . GC2, &c. b. SG Corol. 3. Hence, the distance of the centre of percussion always exceeds the distance of the centre of gravity, and the excess is always Or GO is is always the same constant quantity, wherever the point of suspension S is placed; since the point G, and the bodies A, B, &c. are constant. always reciprocally as SG, that is, GO is less, as SG is greater; and consequently the point rises upwards, and approaches towards the point G, as the n |