186 CELESTIAL MECHANICS. I. iii. 15. 



Supposing the bodies w, wi', 7n!\ to be subject to the 

 force of gravitation only, its action and direction being 

 the same with respect to the whole system, we shall have 



^r-=-^r-==-F— » • "land the equation =:'2mS(i/-rr x^) 



hz 6z^ dz^^ ^ ^^Sx Sy^ 



(n\ becomes S (^_?2w?/— -fswar)> since the quantity _ 



is the same for all the bodies concerned, as well as the 

 force S: and the conditions of the equations, thus trans- 

 formed, may be fulfilled, by putting 



^mx—0, S/wyzzO, and S?wjrzzO. (o) 



The three forces ^mS ^^, '^mS ^-» and Sm^S, -^ parallel 



hx dy dz 



the three axes, which are destroyed by the reaction of the 



^'" ,g' 



fixed point, become, for a similar reason, S^lm,S~^m, 



dx by 



and^^Sm; and these forces compose a force SXm, 

 which is equal to the weight of the body; since (77^)^ + 



(^j^ + (^)^are alwaysrzl, and tlie resulting force is 



expressed by the diagonal of the parallelepiped. 



Scholium 1. The origin of the coordinates, thus con- 

 sidered as the fixed point of the system, is very remarkable 

 for the property of affording an equilibrium of the weight 

 of the whole system, whenever it is simply supported, 

 whatever the angular situation of the system may be. 

 Hence it is called the centre of gravity of the system. Its 

 place is determined by the property, that if we suppose 

 any plane to pass through this point, the sum of the 



