*5 2

Mechanics.

how depend on each other, they will havea Common Centre of Gravity, which will be a

Point

PI. VII.Fig. 2.

F-'g- 3*

Fig- 4-

Base AE, the Centre of Gravity C will be free to de-scend, as not being supported, and consequently theBody must fall.

7. If a Body ABDE, laid on an inclined Plane XY,gravitates in the Direction CI, within the Base A E,that Body will move down the Plane (by Annot,XXVII. i.) but it cannot move over the angular PointA (by Hth of this) it must the.efore descend by Jlidingdown the Plane. If the Base AE, had been so smallthat the Point F had fallen off it, it would have turnedon the Point A in its Motiop, and so have tumbleddown the Plane. But if the Bases AE be supposed infi-nitely small, or a Point, the falling from one Point toanother, will be so momentarieous, that the Body(which then becomes a Circle or Sphere) will descendby rolling down the Plane.

8. To find the Centre of Gravity of a Line , has beenshewn ( Annot. XXIX. 2.) To find that of a Superfi-cies, as of a Triangle , Parabola , &c. this is the Me-thod. Let AG bisect the Base A C of the Triangle AABC, it will also bisect every other Line DE drawn "parallel to the Base; consequently the Centre of Gra-vity of the Triangle will be found somewhere in theLine BG. The Area of the Triangle may be consi-dered, as consisting of an infinite Number of indefi-nitely small Parallelograms DEbaD, each of which is

to be considered as a small Weight, and also as theFluxion of the Area of the Triangle, and so may beexpressed by zyx, (putting BF — x, and FE — y ,) ifthis fluxionary Weight be multiplied by its Velocity x,we shall have 2 yxx for its Momentum.

g. Now put B G = a, and A C = b ; then B G (a) :

A C (£J :: B F (x) : DE = — = 2 y; therefore the

Fluxion of the Weights zy x

b X X

And the Fluxion

3