But it has been shewn that the co-ordinates of the centre of gravity

These three equations determine the motion of the centre of gravity. 152. Thus the centre of gravity moves as if all the bodies of the system were united in that point, and as if all the forces which act on the system were applied to it.

153. If the mutual attraction of the bodies of the system be the only accelerating force acting on these bodies, the three quantities ΣΜΧ, ΣΜΥ, ΣmZ are zero.

Demonstration. - This evidently arises from the law of reaction being equal and contrary to action; for if F be the action that an element of the mass m exercises on an element of the mass m', whatever may be nature of this action, m'F will be the accelerating force with which m is urged by the action of m'; then if f be the mutual distance of m and m', by this action only

m'F (x'

X = -"); Y = m'F (y' - y) : Z = m'F (z' - 2). (23).

f

;

f

;

For the same reasons, the action of m' on m will give

f

mX + m'X' = 0; mY + m'Y' = 0; mZ + m'Z' = 0 ; and as all the bodies of the system, taken two and two, give the same results, therefore generally

Σ.ΜΧ = 0; Σ.ΜΥ = 0; Σ.Ζ = 0.

z=at+b; y = at + b'; z=a"t + b';

in which a, a', a"; b, b', b", are the arbitrary constant quantities intro

duced by the double integration. These

155. Thus the motion of the centre of gravity in the direction of each axis is a straight line, and by the composition of motions it describes a straight line in space; and as the space it moves over increases with the time, its velocity is uniform; for the velocity, being directly as the element of the space, and inversely as the element of the time, is

or

dy

2

√(+)+();

√a2 + a2 + a2.

dt

Thus the velocity is constant, and therefore the motion uniform. 156. These equations are true, even if some of the bodies, by their mutual action, lose a finite quantity of motion in an instant. 157. Thus, it is possible that the whole solar system may be moving in space; a circumstance which can only be ascertained by a comparison of its position with regard to the fixed stars at very distant periods. In consequence of the proportionality of force to velocity, the bodies of the solar system would maintain their relative motions, whether the system were in motion or at rest.

On the Constancy of Areas.

158. If a body propelled by an impulse describe a curve A M B, fig. 42, in consequence of a force of attraction in the point o, that force may be resolved into two, one in the direction of the normal AN,

and the other in the direction of the element of the curve or tan

area MoB.

N

fig. 43.

M

A

gent: the first is balanced by the centrifugal force, the second augments or diminishes the velocity of the body; but the velocity is always such that the areas AoM, MoB, described by the radius vector Ao, are proportional to the time; that is, if the body moves from A to M in the same time that it would move from M to B, the area AoM will be equal to the

If a system of bodies revolve about any point in consequence of an impulse and a force of attraction directed to that point, the sums of their masses respectively multiplied by the areas described by their radii vectores, when projected on the three co-ordinate planes, are proportional to the time. Demonstration. For if we only consider the areas that are projected on the plane

B

0

CD

xoy, fig. 43, the forces in the direc

tion oz, which are perpendicular to that plane, must be zero; hence

Z = 0, Z' = 0, &c.;

and the general equation of the motion of a system of bodies becomes

If the same assumptions be made here as in article 133, namely,

and if these be substituted in the preceding equation, it becomes,

are obtained for the motions of the system with regard to the planes

xoz, yoz. These three equations, together with

are the general equations of the motions of a system of bodies which does not contain a fixed point.

159. When the bodies are independent of foreign forces, and only subject to their reciprocal attraction and to the force at o, the sum of the terms

m {Xy - Yx}+m' {X'y' - Y'x' },

arising from the mutual action of any two bodies in the system, m, m', is zero, by reason of the equality and opposition of action and reaction; and this is true for every such pair as m and m", m' and m", &c. If f be the distance of m from 0, F the force which urges the body m towards that origin, then

X = - F, Y = - F

f

Z=-F

f

are its component forces; and when substituted in the preceding equations, F vanishes; the same may be shown with regard to m', m", &c. Hence the equations of areas are reduced to

As the first members of these equations are the sum of the masses of all the bodies of the system, respectively multiplied by the projec

1

tions of double the areas they describe on the co-ordinate planes, this sum is proportional to the time.

If the centre of gravity be the origin of the co-ordinates, the preceding equations may be expressed thus,

cdt =

c'dt =

c"dt=

Σmm'

{(x'

x) (dy' - dy) — (y' - y) (dx' - dr) }

Ση

Emm' { (z' – z) (dx - dx) - (x – x) (dz' – dz) }

Ση

Emm' { (y' - y) (dz' - dz) - (z' – z) (dy' - dy) }

Ση

So that the principle of areas is reduced to depend on the co-ordinates of the mutual distances of the bodies of the system.

160. These results may be expressed by a diagram, Let m, m', m", fig. 44, &c., be a system of bodies revolving about o, the origin

in an indefinitely small time, represented by dt; and let mon, m'on', &c., be the projections of these areas on the plane roy. Then the equation

Em {xdy - ydx} = cdt,

shows that the sum of the products of twice the area mon by the mass m, twice the area m'on' by the mass m', twice m'on' by the mass m", &c., is proportional to the element of the time in which they are described: whence it follows that the sum of the projections of the areas, each multiplied by the corresponding mass, is proportional to the finite time in which they are described. The other two equations express similar results for the areas projected on the planes

roz, yoz.

161. The constancy of areas is evidently true for any plane whatever, since the position of the co-ordinate planes is arbitrary. The