By a similar mode of proceeding we shall obtain Ꮳ = Σ(Px) ..(vi). These two results (v) and (vi) determine the position of the centre of gravity of the system of particles, which lies in the plane of the particles. III. To find the centre of gravity of a system of particles arranged in any manner in space. Let the system of particles A, B, C... whose weights are P, Q, R... be referred to three lines Ox, Oy, Oz mutually at right angles; let g, be the C.G. of A and B. 9 R 2 2 ... 1 points draw in the plane xOy the lines NM, N2M..........n ̧m ̧, nm....... parallel to Oy meeting Ox in M1, M ̧... If now OM1=x1 MN=y, and similar quantities for each particle, and if x y z be the corresponding quantities for G, the centre of gravity of the system,—we have, considering A and B only at first, P. Ag1 = Q. Bg1; or if we draw lines through A,, g1 parallel to N,N,, we have similarly introducing another particle C, g, being the centre of gravity of A, B, C, and therefore the centre of gravity of P+Q at g, and R at C; (P+Q + R) gn2 = (P+Q) g1n, +R. CN and so on for any number of particles-till we get (P+Q+...) GV=P.z1+Q •≈2+... 2 P.z1+Q.z2+... _Σ (Pz) These three expressions for x y z determine the position of the centre of gravity of the system of particles considered. This includes I. and II. as particular cases. 75. Obs. In the case III. of the preceding article it will in general be convenient to take the lines Ox, Oy, Oz at right angles, but the student will observe that the course of the proof does not require that the lines Ox, Oy, Oz should be inclined at any particular angles: he may then in any par ticular case assume three lines (not in one plane) inclined at any angles which may appear to him most convenient in the case under his consideration;—and a similar remark applies to case II. DEF. The moment of a force with respect to a plane is the product of the force into the distance of its point of application from the plane. If the points of application of two forces are on opposite sides of a given plane, the moments of the forces with respect to that plane will have opposite signs. This must be carefully distinguished from the moment of a force with respect to a point or an axis. Art. 31. COR. 1. We see from the results of Art. 74, that the algebraic sum of the moments of the particles of a system with respect to any plane is equal to the moment of the whole (supposed to be collected at the centre of gravity) with respect to the same plane. From whence follows the conclusion, that if the algebraic sum of the moments of a system taken with respect to any proposed plane be zero, the centre of gravity of the system lies in that plane; and vice versâ, if the centre of gravity of a system lie in a given plane, the algebraic sum of the moments of the particles with respect to that plane is zero,in other words, the sum of the moments of the particles which are on one side of the plane is equal to the sum of the moments of the particles which are on the other side of the plane. -or COR. 2. If we suppose a system to be divided into any number n of particles of equal weights we have the distance of centre of gravity from any planeth the sum of the n distances of all the particles from the same plane. Viewed in this manner, the centre of gravity of a body or system is sometimes called the centre of mean position of the body or system, or the centre of figure. COR. 3. If a system of particles be projected on any plane, the projection of the centre of gravity of the system on that plane will be the centre of gravity of a system of particles in the plane, equal to the former and coincident with the points of projection of the original system. This appears at once from the results of Art. (74), for the values of xyz depend only on the weights of the particles and their distances estimated parallel to Ox, Oy, Oz from the planes y Oz, zОx, x Oy severally. 76. Centre of parallel forces. If in any of the cases of Art. (74), A, B, C... be the point of application of a system of parallel forces P, Q, R,... the method pursued in that article will lead to formulæ for the co-ordinates of the point of application of the resultant of such a system of parallel forces, viz. These results are algebraically true whether the forces act all in the same direction or not-and we may interpret them as stating that the resultant of a system of parallel forces is = (P) acting at a point whose co-ordinates are given by equation (i). If however Σ (P) =0, and the expressions Σ (Px), Σ (Py), Σ(P2) do not each = 0 also, the system will be equivalent to a couple which does not admit of being represented by a single resultant force, Art. (30). 77. The position of the centre of gravity of a body or a system of particles depends (as we have seen, Art. 74) only on two things; (i), the form of the body, or, in other words, the arrangement of the particles of the system; and (ii), the relative density of the different parts. Formulæ have been obtained in Art. 74, by which the centre of gravity of any system of particles whose relative weights and position are known, may be found; and we have seen in Cor. 1, Art. 70, that a body may be considered as a particle placed at the centre of gravity of the body, so that if the centres of gravity of the several bodies composing a system be known, we are enabled to find the centre of gravity of the system, and the problem assumes a general character. The determination however of the centre of gravity of a body (either a continuous solid body, or a surface regarded as a lamina of matter of indefinitely small thickness) will in general require the aid of the Integral Calculus. Obs. Cases will not unfrequently arise in which the position of the centre of gravity can be assigned from geometrical considerations such as the following, which are suggested for the consideration of the student. 1o. If in any body or system a plane can be found which divides the body into two parts which are symmetrical with respect to the plane on opposite sides of it, the centre of gravity of the body must lie in that plane. |