1 74
Mechanics,
PlatelX.Fig. 2.
by putting C a == a, C b — b, and C m—d-, then
~ — — d. Suppose the Moment the Rod C D/\ -j- £>
began to descend, the Bodies A and B were to be dis-engaged from it in such a Manner as to descend upon itfreely; then will they descend (not in the Curves apand b q, but) in the Tangents a c and b d, describingthe Spaces ac, b d, which will be as the Inceptive Ve-locities Ca, C b; and of course when the Rod comesinto the Situation C d, the Bodies will be at c and d ;it being in this Cafe C a : C b : : a c : b d:: C c : C d.And mn will be the Line described by the commonCentre of Gravity.
27. At c and d, let us suppose the Bodies again fix’dto the Rod, they will then constitute a Compound Pen-dulum, which will now vibrate forwards and backwardsthrough the Angle dCr, the Body A describing theArch ces, and B the Arch dfr, and the commonCentre of Gravity r the Arch n 0 I. As this Pendulumvibrates, the Velocities of the Bodies will be everywhere as their Distances, and greatest of all in thePerpendicular Situation Cef: Where the Velocities ofthe Points c, 0, f, may still be represented by a, d,and b.
28. Now this Compound Pendulum is isochronous to(/’. e. vibrates in the same Time with) some simplePendulum of a determinate Length P Q. If then inthe Compound Pendulum C d there be taken C x — P Q,the Point x is called the Centre of Oscillation in thisCompound Pendulum ; for if the Bodies A and B werethere fixed, the Times of Vibration would be the famethen as now : To determine this Point therefore, is todetermine the Time of Vibration in any Compound Pen-dulum whatsoever.
29. In order to this ; let us once more suppose theMoment the Bodies A and B come to the Points e and sin the Perpendicular, they were disengaged from theRod, and were to ascend at the End of a String sepa-rately, with the Velocities they had acquired by theirDescent in the Points e and f. Then the Altitudes i gand k h, to which thev will ascend above the Horizon-tal Line e i and f h, will be as the Squares of the Ve-
2 locities