106 THEORY OF NEWTONIAN FORCES. [PT. I. CH. II. 



is in the direction of the normal to the surface. But under these 

 conditions the particle is in equilibrium. 



In like manner we may show that if the forces X lt Y lt Z l} act 

 upon the particle 1, X 2) Y 2 , Z 2) upon the particle 2, etc., the 

 condition of equilibrium is 



( 1 2) XM\ + Fifyx + Z^Z, + Xa + K% 2 + ZJlZt ...... + Z n Sz n = 0, 



where the displacements satisfy 



(13) 



Multiplying the equations (13) respectively by \ lt X 2 , . . . X^, and 

 adding to (12) we have 



Of the 3n quantities 8x l} Sz n , only 3^-A; are arbitrary, 



we may however determine the k multipliers X so that the coeffi- 

 cients of the k other S's vanish, then the coefficients of the 3n k 

 arbitrary S's must vanish, so that we get the 3n equations 



