Properties of the Centre of Gravity. 71 



117. Also, since several forces applied at the same point, 

 reduce themselves, by the preceding principles, to a single one, 

 we infer generally, that whatever be the number, direction, and mag- 

 nitude of the forces which are applied at the centre of gravity of a 

 system of bodies ; 



(1). All parts of this system will have the same velocity ; 



(2). This velocity will be in the direction of the resultant of all 

 the applied forces ; 



(3). It will be equal to the quantity of motion, which this result- 

 ant represents, divided by the entire mass of the system. 



118. Whence we conclude, that while the forces which act upon 

 a body, are capable of being reduced to a single one, the direction of 

 which passes through the centre of gravity, this body will not turn 

 about the centre of gravity. 



119. But if the forces which act upon a body cannot be re- 

 duced to a single one, or on the supposition that they admit of 

 being so reduced, if the direction does not pass through the cen- 

 tre of gravity, all the parts of the system will not have a common 

 motion. Nevertheless the centre of gravity will move in the 

 same manner as if all the forces were applied directly at this 

 point, as we now propose at show. 



120. Let us in the first place suppose three bodies m, n, o, Fi<y 

 moving in parallel lines AA", BB", CC", (situated in the same or 



in different planes,) and with velocities represented by the lines 

 AA", BB", CC", respectively, the motion of each being uniform. 

 Let us suppose also, that G is the centre of gravity of these 

 bodies, when they are in A, B,C-, and that G" is their centre of 

 gravity, when they are in A", B", C", where they will arrive in 

 the same time, since their velocities are represented by AA' f , 

 BB", CC" ; joining GG", I say that this line will be parallel to 

 AA", BB", &c., and ti<at it will represent the course described 

 by the centre of gravity during the supposed motion of the bodies 

 m, n, o, and that it will be described uniformly. 



(1). It is evident that the course described by the centre of 

 gravity will he parallel to the lines AA", BB", SLC. ; for at what- 

 ever point we suppose it at any instant, if we imagine a plane pass- 

 ing through it, the sum of the moments with respect to this plane 



