Mechanics.

*75

Iocities at e and f ; that is, as a a and b b ; as is evi-dent from what has been elsewhere shewn (Annotat.XXVI. 2.3.)

30. The Bodies A and B will in this Cafe arrive tothe Points g and h before their Motion will be destroy’d.And it is plain that eg is less than ei or e s, because theBody A is retarded in coming from C to r by theBody B. Also f h is greater than df or f r, because Bhas its Motion quickened by the Body A. And aswhen in passing from d to f, the perpendicular Spacesdescended through are as the Velocities singly, and theMomenta as the V elocities and Masses of Matter con-jointly ; so on the other Side, in passing fromy and eto h and g, the perpendicular Spaces ascended throughareas the Squares of the initial Velocities, the Momentawill be as the Masses of Matter, and those perpendi-cular Spaces conjointly ; that is, as A a a and B b b.

31. Now in all Cafes, the Sum of the Momenta di-vided by the Sum of the Masses gives a Quotient ex-pressing the Velocity, or Ascent or Descent ot the Centre

of Gravity. (See Art. 26.) Therefore- - ^ ^ —-

— d— the Ascent of the Centre of Gravity ; for theMomentum of this Centre of Gravity is on both Sidesthe Perpendicular C f the fame at equal Distances, andtherefore it will always ascend or descend through thesame perpendicular Heights.

32. Having thus obtained the Expression of the De-scent of the Centre of Gravity, we have that for theCentre x of Oscillation of course, for we have As C n :

~ Aaa-\-Bbb A a a x -j- B b b x

C x \\d: x a+~B : ."aT+BT “ =

Descent of the Centre of Oscillation. But the Descentis equal to the Ascent; for the Point x is actuated andmoved in the fame Manner as the Body Q, which inone Oscillation descends and ascends through equal per-pendicular Spaces, and those Spaces are always as theSquares of the Velocities in the lowest Point; and there-fore for the Point *, it will be as C x- —xx. Whence

, AaaxA-Bbbx , ,

we have --—- , —- = x x : and therefore

A d + B d

A a a