when m is in equilibrio. A similar equation must exist for each coordinate plane when m is in a state of equilibrium with regard to each axis, therefore also m (xdz – zdr) = 0, m (ydz – zdy) = 0. The same may be proved for every body in the system, consequently when the whole is in equilibrio on the point o Em(rdy – ydr)=0 Em (rdz – ~d)=0 Em (ydz - zdy) = 0. (15). 133. This property may be expressed by means of virtual velocities, namely, that a system of bodies will be at rest, if the sum of the products of their momenta by the elements of their directions be zero, or by article 125 Σm Fds=0. Since the mutual distances of the parts of the system are invariable, if the whole system be supposed to be turned by an indefinitely small angle about the axis oz, all the co-ordinates z', z", &c., will be invariable. If d be any arbitrary variation, and if Sy' x'd; then f being the mutual distance of the bodies m and m' whose co-ordinates are x, y, z; x', y', z', there will arise dƒ=d√(x' — x)2 + (y' — y)2 + (z' — 2)2 = } { (x'—x) (y' − y) dã — (y' — y) (x' − x) dw } = 0. So that the values assumed for dr, dy, dr', dy' are not incompatible with the invariability of the system. It is therefore a permissible assumption. Now if s be the direction of the force acting on m, its variation is since z is constant; and substituting the preceding values of dr, dy, the result is In the same manner with regard to the body m' In fact, if X, Y, Z be the components of the force F in the direction. of the three axes, it is evident that parallel to the axis of a to turn the system round that of z, and Σm Fr ds that of the forces parallel to the axis of y to do the same, Sy but estimated in the contrary direction;—and it is evident that the forces parallel to z have no effect to turn the system round z. There fore the equation ΣmF (3 ds бах x =0, expresses that the sum of бу the moments of rotation of the whole system relative to the axis of z must vanish, that the equilibrium of the system may subsist. And the same being true for the other rectangular axes (whose positions are arbitrary), there results this general theorem, viz., that in order that a system of bodies may be in equilibro upon a point, the sum of the moments of rotation of all the forces that act on it must vanish when estimated parallel to any three rectangular co-ordinates. 134. These equations are sufficient to ensure the equilibrium of the system when o is a fixed point; but if o, the point about which it rotates, be not fixed, the system, as well as the origin o, may be car ried forward in space by a motion of translation at the same time that the system rotates about o, like the earth, which revolves about the sun at same time that it turns on its axis. In this case it is not only necessary for the equilibrium of the system that its rotatory pressure should be zero, but also that the forces which cause the translation when resolved in the direction of the axis or, oy, oz, should be zero for each axis separately. On the Centre of Gravity. 135. If the bodies m, m', m", &c., be only acted on by gravity, its effect would be the same on all of them, and its direction may be considered the same also; hence are the same in this case for all the bodies, so that the equations of or, if X, Y, Z, be considered as the components of gravity in the three co-ordinate axes by article 133 It is evident that these equations will be zero, whatever the direction of gravity may be, if Σmx = 0, Σmy = 0, Σmz = 0. (18). components of the force Now since F, F ds of gravity in the three co-ordinates ox, oy, oz, are the forces which translate the system parallel to these axes. But 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. 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 m" fig. 39. 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", hence m" 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 x x′ x", &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 7, y, z, it is a single point. |