102 CELESTIAL MECHANICS. I. 1. S. 

 d S oS X CL 



that -7— or-^=i . If there be a second force iS', 



^x &x s 



and / be the distance of 31 from any fixed point in its 

 direction, we have in a similar manner S\ r- for the por- 

 tion of this force acting in the direction of x ; and employ- 

 ing the characteristic 2 for the sum of all the forces thus 



determined, we have 'L S. j- for the whole force in the 



direction of x. Now if F be the result of all the forces 

 S, S', S^\ . . . thus combined, and u the distance of any 



point in its direction from M, we have F. -r- for the por- 

 tion of this force, which acts in the direction of x, and 

 which must be equal to 2 S. -r-, by the supposition : and 



by comparing, in the same manner, the forces in the direc- 



^'u ^s ^'u 



lions y and z, we obtain F. -.r- =: X S. c-~, and F. -^r 



^ ai/ &y &z 



z= 2 *S. -y- ; and then, adding these partial variations, we 



obtain V^uzzXS.h, an equation which may be said to con- 

 tain the three formeK|||^because, since the variations are 

 perfectly arbitrary, we may make any two of them vanish, 

 and the third will remain alone on both sides of the 

 equation. 



251. Corollary 1. When the point re- 

 mains in equihbrium, the sum of the products 

 of each force, multipHed by the elementary- 

 variation of its distance, is equal to nothing. 



Or XS.^zzO ; since F=0. (h) 



