122

Mechanics.

Centre of Oscillation E in the Rod, is alsothe Centre of PercuJJion , or that Point inwhich the Force of the Stroke is the greatestpossible (XXIX).

therefore the Times of Descent thro’ the Lengths ofthe Pendulums are equal. But Bodies descending thro’unequal Spaces in equal Times are impcll’d by Powersthat are as the Spaces described, that is, the Powers ofGravity are as the Lengths of the Pendulums. See Annot.XXVII. Art. 2 and 3.

PI. VI. (XXIX) I. If AB be an uniform Rod vibratingFig. 6. about the Point A, the Velocities of every Point D, C,

B, will be as the Arches D E, C F, BG described inthe same Time ; but those Arches are as AD, AC,AB; consequently the Velocity of every Point is propor-tional to its Distance from the Center of Motion A.

2. Therefore if 5 represent the Length of theLine, it will represent also the Sum of all the Ve-locities, of which x will be the Fluxion ; therefore jxx will be the Fluxion of the Momenta, and so

the Fluent — will be the Sum of all the Momenta , ;

2

which divided by the Quantity of Matter at, will give

— for an Expression of all the Velocities on each Side

the common Center of Gravity of all the Points, which 'therefore is in the middle Point of the Line A B. ;

3. In regard of striking an Object, the Momentum ofeach Particle will be increased in Proportion to its Ve-locity and Distance from the Center A, and this is call’dthe Force of the Particle ; therefore if the Fluxion ofthe Momenta x x be multiplied by the Velocity x, the iProduct xxx will be the Fluxion of the Forces of all

the Particles : The Fluent of which — will be the Sum

3

of all the Forces, which divided by the Sum of all the

x 1

Momenta — will give \ x = Distance of the Center of2

Forces from. the Point A. In this Center C, therefore,

th,e