if o be a fixed point, its reaction would destroy these forces.

are the sides; therefore these three compose one resulting force equal to F. Em. This resulting force is the weight of the system which is thus resisted or supported by the reaction of the fixed point o.

136. The point o round which the system is in equilibrio, is the centre of gravity of the system, and if that point be supported, the whole will be in equilibrio.

On the Position and Properties of the Centre of Gravity.

m'

m"

fig. 39.

137. It appears from the equations (18), that if any plane passes through the centre of gravity of a system of bodies, the sum of the products of the mass of each body by its distance from that plane is zero. For, since the axes of the co-ordinates are arbitrary, any one of them, as x o x', fig. 39, may be assumed to be the section of the plane in question, the centre of gravity of the system of bodies m, m', &c., being in o. If the perpendiculars ma, m'b, &c., be drawn from each body on the plane xox', the product of the mass m by the distance ma plus the product of m' by m'b plus, &c.,

m

m

d

m

must be zero; or, representing the distances by z, z', z", &c., then mz + m'z — m” z" + m” z′′ + &c. = 0 ;

or, according to the usual notation,

Σ.mz = 0.

And the same property exists for the other two co-ordinate planes Since the position of the co-ordinate planes is arbitrary, the property

F

obtains for every set of co-ordinate planes having their origin in o. It is clear that if the distances ma, m'b, &c., be positive on one side of the plane, those on the other side must be negative, otherwise the sum of the products could not be zero.

138. When the centre of gravity is not in the origin of the coordinates, it may be found if the distances of the bodies m, m', m”, &c., from the origin and from each other be known.

m

IM

a
C

fig. 40.

m

pp A

N

explained,

but

hence

m.ma + m'. m'b m".m"d+ &c. = 0;

maoA

op; m'boA - op', &c. &c.,

m (oA — op) + m2 (oA — op') + &c. = 0;

or if Ao be represented by 7, and op op' op", &c., by xxx", &c.,

Thus, if the masses of the bodies and their respective distances from the origin of the co-ordinates be known, this equation will give the distance of the centre of gravity from the plane yoz. In the same manner its distances from the other two co-ordinate planes are found to be

139. Thus, because the centre of gravity is determined by its

three co-ordinates x, y, z, it is a single point.

(Emx)2 + (Σmy)2 + (Σmz)2
(Em)2

Σm(x2+y2+~2) _ Σmm'{(x'−x)2+(y'-y)2 + (z'−z)°}
Ση

(Σm) 2

The last term of the second member is the sum of all the products similar to those under Σ when all the bodies of the system are taken in pairs.

141. It is easy to show that the two preceding values of ♬2+ÿ2+~ are identical, or that

or

(Emx) Em. Emx2 - Emm' (x'
=

Were there are only two planets, then

Σmm+m', Σmx = mx + m'x', Emm' = mm' ;

consequently

(Emx)=(mx + m'x')2 = m2x2 + m22x22 + 2mm'xx'.

With regard to the second member

Σm. Emr2=(m+m') (mx2+m'x'2)=m2x2+m22x2+mm'x2+mm'x2', and Σmm' (x' — x)2 == mm'x'2 + mm'x2 2mm'xx';

consequently

Em. Emx2 - Emm' (x' — x)2 = m2x2 + m22x22 + 2mm'xx' = (Σmx)2. This will be the case whatever the number of planets may be; and as the equations in question are symmetrical with regard to x, y, and z, their second members are identical.

Thus the distance of the centre of gravity from a given point may be found by means of the distances of the different points of the system from this point, and of their mutual distances.

142. By estimating the distance of the centre of gravity from any three fixed points, its position in space will be determined.

Equilibrium of a Solid Body.

143. If the bodies m, m', m", &c., be indefinitely small, infinite in number, and permanently united together, they will form a solid mass, whose equilibrium may be determined by the preceding equations.

For if x, y, z, be the co-ordinates of any one of its indefinitely small particles dm, and X, Y, Z, the forces urging it in the direction of these axes, the equations of its equilibrium will be

SXdm=0 Ydm=0 Zdm=0

ƒ (Xy—Yx) dm = 0; ƒ(Xz — Zx) dm = 0; f(Zy – Y≈) dm=0. The three first are the equations of translation, which are destroyed when the centre of gravity is a fixed point; and the last three are the sums of the rotatory pressures.

CHAPTER IV.

MOTION OF A SYSTEM OF BODIES.

144. It is known by observation, that the relative motions of a system of bodies, are entirely independent of any motion common to the whole; hence it is impossible to judge from appearances alone, of the absolute motions of a system of bodies of which we form a part; the knowledge of the true system of the world was retarded, from the difficulty of comprehending the relative motions of projectiles on the earth, which has the double motion of rotation and revolution. But all the motions of the solar system, determined according to this law, are verified by observation.

By article 117, the equation of the motion of a body only differs from that of a particle, by the mass; hence, if only one body be considered, of which m is the mass, the motion of its centre of gravity will be determined from equation (6), which in this case becomes

dy) by + m {Z - dz}
} sy

dt2

dte

dt

= 0.

A similar equation may be found for each body in the system, and one condition to be fulfilled is, that the sum of all such equations must be zero;—hence the general equation of a system of bodies is

are the sums of the products of each mass by its corresponding component force, for

ΣmX = mx + m'X' + m"X" + &c. ;

are the sums of the products of each mass, by the second increments

of the space respectively described by them, in an element of time in the direction of each axis, since