OF THE EQUILIBRIUM OF A SYSTEM. 183 



tion with the equation (k), we shall have ^>^du—'2pdf-\- 

 ^RBr ; whence it will be easy to infer the reciprocal ac- 

 tions of the bodies m, m\ . . . , as well as the pressures 

 — R, — R', . . . , which they exert on the surfaces to which 

 they are confined. 



§ 15. Conditions of equilibrium for a system, of which 

 all the points are united in an invariable manner. Centre 

 of gravity : mode of determining its position with respect 

 to three planes or three given points, P. 42. 



307. Theorem. The forces acting on any 

 system of bodies in equilibrium being referred 

 to three orthogonal directions, the sum of all the 

 forces acting in each direction must vanish, as 

 well as the sum of the rotatory pressures with 

 respect to axes in each of the three directions. 



If all the bodies of a given system be invariably united 

 to each other, its position will be determined by that of 

 any three points belonging to it, which are not in a right 

 line : now the position of each of these points depends on 

 three coordinates, so that nine different distances are 

 comprehended in their equations: but since the three 

 distances of the points are given, they reduce the number 

 of independent quantities to six, which will afford as many 

 arbitrary variations : and by supposing the coefficients of 

 these to vanish, we shall obtain six equations, which will 

 include all the conditions of the equilibrium. 



For this purpose, we may suppose x, y, z, to be the 

 coordinates of m; x, y\ z\ those of m', and x"y y", z'\ 

 those of m", . . . ; we shall then have 



