OF THE EQUILIBRIUM OF A SYSTEM. 181 



that the sum of these last forces must be equal to nothing, 

 since the system is at liberty to move in the respective di- 

 rections, [unless each point were held at rest by equal and 

 opposite forces, so that the sums of the opposite forces 

 must be equal for all the points, and all these forces will 

 vanish,] producing neither pressure on the given curves, 

 nor reaction between the bodies, so that they may be ex- 

 cluded from the equation, and the forces ^, jp', 'j^' must be 

 in equilibrium without them, or in other words — 'p^ —p', 

 -—p", . . . together with q, q\ g", . , . , must afford an equi- 

 librium among themselves. Now, if ^i, ^i', ... be the va- 

 riations of the lines of direction of the forces T, T\ T",,„, 

 we shall have, from the equation (k)fi=^(q^-p) ^f+^T^i; 

 but the system being supposed to remain at rest in conse- 

 quence of the forces q, q\ . . . , without any action upon 

 the curves or surfaces, the equation (k) gives us also 0= 

 ^qdf: consequently OziSp^— ST^z. But in the condi- 

 tions of the problem ^i=0, S^'=0, ... , the variations be- 

 ing confined to the curves, so that we have finally 0=2/?^, 

 whence it follows, that with the conditions of the connexion 

 of the system, ^mS^szuO, as before. 



[Scholium. The object of the second part of the de- 

 monstration is to prove that ifp, p', p", . . . , represent not 

 the reciprocal actions, but the total forces exerted on each 

 body, exclusive of the pressure of the surfaces, these 

 forces may be decomposed so as to afford forces equivalent 

 to the reciprocal actions of the respective bodies, and that 

 the remaining portions of the forces, as well as these reci- 

 procal actions, will balance each other, in the case of equi- 

 librium, according to the terms of the proposition]. 



306. Corollary. The converse of this 

 proposition is equally true, and whenever the 



