Properties of the Centre of Gravity. 75 



three motions are afterward reduced to one (which may be done 

 since they are all applied at the same point,) we shall have the 72 * 

 course of the centre of gravity in a single line. Also, as the 

 elements here employed, are simply the forces themselves which 

 the bodies have parallel to the three co-ordinates, and as the 

 single force of the centre of gravity is thus found to be composed of 

 the resultant forces parallel respectively to these given lines, it 

 cannot but be equal and parallel to the resultant of all the forces 

 applied to the bodies in question; hence, whatever be the directions 

 and magnitudes of the forces applied to different parts of* a system of 

 bodies, the centre of gravity moves always, or tends to move, in the 

 same manner, as if the forces in question were all applied directly at 

 this point* 



125. In the foregoing article, we have said that we may al- 

 ways decompose the velocity of each body into three others, 

 parallel respectively to three lines whose position is given. If 

 ihe direction of one of the bodies, however, be parallel to the 

 plane of two of the three assumed lines, or if it be parallel to one 

 of these lines, it might seem that, in the first case, it would not 

 admit of being decomposed, except into two forces, parallel to 

 two of the three given lines ; and that in the second case, no 

 decomposition whatever could take place into forces parallel to 

 the two other lines. Notwithstanding this apparent difficulty, 

 the proposition is true universally. We see, for example, that 



so long as the line AB is not parallel to either of the lines XZ, Fig. 54. 

 XT, we can always decompose the force represented by AB 

 into two others, AC, AD, parallel to these two lines respectively ; 

 but we perceive, at the same time, that the more AB approaches 

 to a parallelism with XT, the more the force AD diminishes ; so 

 that it becomes zero, when AB is parallel to XT. There is not, 

 therefore, in this case, the less propriety in supposing a decom- 

 position into two forces, because one of them is zero. For a like 

 reason, we may, in the same case, suppose a decomposition into 

 three forces, parallel to three given lines XT, XZ, XY, two of 

 which are equal respectively to zero. 



126. From what we have now said, taken in connexion with 

 that of article 122, we infer, that the centre of gravity of a system 

 <of bodies will remain at rest, if, each of the forces applied to the seve' 



