74 Statics. 



nus the sum of the quantities of motion of those that move in tfa 

 opposite direction, divided by the sum of the masses. 



121. If any one of the bodies be at rest, the velocity of this 

 body will be zero, and the quantity of motion also will be 

 zero. Thus it will disappear from the numerator of the frac- 

 tion which expresses the velocity of the centre of gravity ; but 

 the denominator, remaining unchanged, will in every case be 

 the sum of all the masses. 



122. If the sum of the quantities of motion of the bodies 

 which move in one direction, be equal to the sum of the quan- 

 tities of motion of those moving in the opposite direction, the 

 numerator of the fraction which expresses the velocity of the 

 centre of gravity will be zero. This centre of gravity, there- 

 fore, will be at rest. Accordingly, whatever be the parallel 

 motions of several bodies, their common centre of gravity will 

 remain at rest, when the sum of the quantities of motion of those 

 that move in one direction is equal to the sum of the quantities 

 of motion of those that move in the opposite direction. 



28. 1 23. Since the quantities of motion represent the forces ; and 



the resultant of any number of parallel forces is equal to the 

 sum of those which act, or tend to act, in one direction, minus the 



50, sum of those which act, or tend to act, in the opposite direction ; 

 we conclude, that if any number of parallel forces are applied to 

 different parts of a system of bodies, the centre of gravity of this sys- 

 tem will move as if the forces in question were all applied directly 

 at this point. 



124. Let there be any number of bodies moving according to 

 any given straight lines. If we imagine three rectangular co-ordi- 

 nates, we may always decompose the velocity of each body into 

 73. three other velocities, parallel respectively to these three lines. 

 Now it follows from what we have just said, that the motion of 

 the centre of gravity, in virtue of the motions parallel to one of 

 these lines, will be parallel to this same line ; it will also be uni- 

 form, and equal to the sum of the quantities of motion, (estimated 

 parallel to this line) divided by the sum of the masses. If 

 therefore we suppose that the motion of the centre of gravity, 

 parallel to each of these lines, is thus determined, and that these 



