Page:Mécanique céleste Vol 1.djvu/112

82 may put their coefficients equal to nothing, which will give an equal number of equations, by means of which we may determine the functions $$\lambda, \lambda',$$ &c.

If we then compare this equation with the equation (k) [110], we shall have

2.x.5M = 2.p.(5/+2.i?.(5r;

whence it will be easy to deduce the reciprocal actions of the bodies m, m', &c., and the pressures — R, — E, &c., which they exert against the surfaces, upon which they are forced to remain.

15. If all the bodies of a system are firmly attached together, its position may be determined by any three of its points, which are not in the same right line. The position of each of these points depends on three co-ordinates, which produce nine indeterminate quantities; but the mutual distances of the three points being given and invariable, we may, by means of them, reduce these quantities to six others, which, substituted in the equation (l) [110"], will introduce six arbitrary variations; putting their coefficients equal to nothing, we shall have six equations, which will contain all the conditions of the equilibrium of the system; we shall now develop these equations.

Let x, y, z be the co-ordinates of m; x', y', z' those of m' ; x′′, y′′, sf′′ those of m′′, &c.; we shall have

/ = xTJ^ - xf + (jj -yf + {z' -zf ',

/' _ / (x"-.xf+{if — yj + {^' — zf ;

/" = V/ (a;" ^ xj + (/ - 'i/f + (2:"- ^)^ ; &c.

If we suppose

5a;==5ar'= 5a;" = &c. ; 6y=iS'i/ = Sy"= &c. ; 8z = S2f = 5zf' = kc.;

changing x', y' ,z' into x′′, y′′, z′′, &c. The variation of f [109b] is
 * (53a) The value / is the same as in [109a]; f', f′′, &c. are of the same form,

8f=A{^x' — Sx)-{-B{Sy'—Sy)--C{S::f — Sz), which, by means of the equations [119], becomes 5/= 0. In like manner we get Sf = 0, Sf" = 0, &z;c. as above.