Σm(x2+y2+~2) _ Σmm'{(x'—x)2+(y'−y)2 + (z'—z)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 x2+y2+z2 are identical, or that Were there are only two planets, then Σm = m + m', Σmx = mx + m'x', Σmm' = mm' ; consequently (Emx) (mx + m'x')2 = m2x2 + m2x2 + 2mm'xx'. With regard to the second member Σm. Emr2=(m+m') (m.x2+m'x'2)=m2x2+m2x2 +mm'x2+mm'x2', and Emm' (x' - x)2 == mm'x'2 + mm'x2 2mm'xx'; consequently Em. Emx2 - Emm' (x' — x)2 = m2x2 + m22x22 + 2mm'xx' = (Emx). This will be the case whatever the number of planets may be; and as the equations in question are symmetrical with regard to r, 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 fYdm=0 fZdm = 0 f(Xy-Yx) dm = 0; ƒ(Xz — Zx) dm = 0; f(Zy – Yz) 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 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 com 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 From this equation all the motions of the solar system are directly obtained. 145. If the forces be invariably supposed to have the same intensity at equal distances from the points to which they are directed, and to vary in some ratio of that distance, all the principles of motion that have been derived from the general equation (6), may be obtained from this, provided the sum of the masses be employed instead of the particle. 146. For example, if the equation, in article 74, be multiplied by Em, its finite value is found to be ΣmV2 = C + 2Σ fm (Xdx + Ydy + Zdz). This is the Living Force or Impetus of a system, which is the sum of the masses into the square of their respective velocities, and is analogous to the equation relating to a particle. V2 C + 2v, 147. When the motion of the system changes by insensible degrees, and is subject to the action of accelerating forces, the sum of the indefinitely small increments of the impetus is the same, whatever be the path of the bodies, provided that the points of departure and arrival be the same. 148. When there is a primitive impulse without accelerating forces, the impetus is constant. 149. Impetus is the true measure of labour; for if a weight be raised ten feet, it will require four times the labour to raise an equal weight forty feet. If both these weights be allowed to descend freely by their gravitation, at the end of their fall their velocities will be as 1 to 2; that is, as the square roots of their heights. But the effects produced will be as their masses into the heights from whence they fell, or as their masses into 1 and 4; but these are the squares of the velocities, hence the impetus is the mass into the square of the velocity. Thus the impetus is the true measure of the labour employed to raise the weights, and of the effects of their descent, and is entirely independent of time. 150. The principle of least action for a particle was shown, in article 80, to be expressed by feds = 0, when the extreme points of its path are fixed; hence, for a system of bodies, it is Thus the sum of the living forces of a system of bodies is a minimum, during the time that it takes to pass from one position to another. If the bodies be not urged by accelerating forces, the impetus of the system during a given time, is proportional to that time, therefore the system moves from one given position to another, in the shortest time possible which is the principle of least action in a system of bodies. On the Motion of the Centre of Gravity of a System of Bodies. 151. In a system of bodies the common centre of gravity of the whole either remains at rest or moves uniformly in a straight line, as if all the bodies of the system were united in that point, and the concentrated forces of the system applied to it. Demonstration. These properties are derived from the general equation (21) by considering that, if the centre of gravity of the system be moved, each body will have a corresponding and equal motion independent of any motions the bodies may have among themselves: hence each of the virtual velocities dr, dy, dz, will be increased by the virtual velocity of the centre of gravity resolved in the direction of the axes; so that they become dx + di, dy + dÿ, dz + dz: thus the equation of the motion of a system of bodies is increased arising from the consideration of the centre of gravity. If the system be free and unconnected with bodies foreign to it, the virtual velocity of the centre of gravity, is independent of the connexion of the bodies of the system with each other; therefore di, 87, 8 may each be zero, whatever the virtual velocity of the bodies themselves may be; hence |