Page:Somerville Mechanism of the heavens.djvu/143

Chap II.] Ckmp. ni.] A SYSTEM OF BODIES. 63 In the same manner with regard to the hody m' and to on ; and thus the equation 2,rnFh = becomes It follows, from the same reasoning, that In fact, if X, Y, Z be the components of the force F in the direction of the three axes, it is evident that X = fJ.; Y=Fii; Z =: f11; is iy iz and these equations become 2my.X — 2mjr. Y = 2m2.X-2wx.Z = (16). 2m».Y - I,my. Z^O But 2,^^y — expresses the sum of the moments of the forces ox parallel to the axis of x to turn the system round that of z, and is SmFx — that of the forces parallel to the axis of y to do the same, iy but estimated in the contrary direction ; — and it is evident that the forces parallel to z have no effect to turn the system round x. There- fore the equation 2mF ( y — — x — |=0, expresses that the sum of Sjp iyj 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 posi- tions are arbitrary), Uiere results tliis 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 wlien estimated parallel to any three rectangular co-ordinates. 134. These equations are suflicient to ensure the equilibrium of the system when o is a fixed point ; but if o, the point about which it ro- tates, be not fixed, the system, as well aa the origin o, may be cai^