/ 



OF THE EQUILIBRIUM OF A SYSTEM. l79 



intersection, the curve to which the motion of m is con- 

 fined, and r, / the hnes perpendicular to these surfaces, 

 we shall have, from the equation 0:=.1.S^s -\- ll'^r -\r R^r 

 (d) (253), OzzmS^s +p^J-\-p'^J' + . . . + i^Sr + m/. In 

 the same manner m' may be considered as a point held in 

 equilibrium by means of the force m!S\ together with the 

 actions of the bodies in, m"^ , » . , and the reactions of the 

 surfaces, which may be called R' and R". If, then, the 

 variation of s' be called ^s\ that of/, taken with regard to 

 this variation, and supposing in to be fixed, ^^Jy that of 

 f", supposing m" fixed, ^,f", and the variations in the 

 directions of R" and R" be ^r" and ^r", we shall have, 

 for the equilibrium of m', O^zm'S'ds -hpdj+p'%f' + -- - 

 '\-R'Zr"-irR"cr"': and the rest of the points will aOoid 

 similar variations, which we may add together, observing 

 that for the total variations, ^f-^J-\-^J> ^f'=Kf + K 

 /'., . .; each distance being liable to two partial variations, 

 one at each end. We shall thus obtain 



OzzXmS^s + Xjy^f+'ER^r. (k) 

 In estimating the forces acting on, each point m, w". . . , it 

 is obvious that we may either consider any number of dif- 

 ferent forces separately multiplied by the respective varia- 

 tions of their distances, or consider the whole as combined, 

 for each body, into a single result, by the equation (a) 

 VBu=XS^s (260). 



If the bodies are united at fixed distances from each 

 other, the lines /,/',/''•.., becoming constant, this con- 

 dition may be expressed by making ^/=:0, ^'=0, ^f 

 = . . . The variations of the coordinates, comprehended 

 in the equation (k), may be subjected to this condition, and 

 then the forces p, expressing the reciprocal actions of the 

 bodies, will no longer be concerned in it : we may also 



N 2 



