Properties of the Centre of Gravity. 73 



and for the same reason, when they are at A', B', C', 

 m X AA' + n X BB 1 o X CC' = (m + + o) X GG'. 



Now since ./?.#', BB', CC', are described uniformly in the 

 same time, these spaces must be as the velocities, AA", BB" 9 26. 

 CC", consequently 



AA" : BB" : : AA' : BB', AA" : CC" n AA' i CO, 



M' x BB" rr/ AA' x CC" 

 which give BB' = -^ , Uf/ = -^ . 



Substituting these values in the last of the above equations, we- 

 shall have 



or, by making the denominator to disappear, 



(m X AA" + n X #B" o X CC") X 



= (w + w -I: o) X GG' X AA''. 

 This equation divided by that in which GG" enters, gives 



A A' = or ^^ X GG" = GG' X A A", 



CrCr 



from which we have 



GG" : GG' : : AA" : AA', 

 which was proposed to be demonstrated. 



We remark that the equation in which GG" enters, gives 



GG" = m x AA '' + nx BB " x cc " 



m + n -}- o 



Now the lines AA", BB",CC", GG", are the velocities respective- 

 ly of the bodies m, n, o, and of the centre of gravity G ; conse- 

 quently m X AA", n X BB", &LC., are the quantities of motion 

 respectively. Accordingly, since the reasoning we have pursued 

 does not depend in any degree upon the number of bodies, we 

 infer, as a general conclusion, 



(1). That if any number of bodies describe parallel Enes, the cen- 

 tre of gravity describes a line parallel to them ; 



(2). That the velocity of the centre of gravity is equal to the sum 

 of the quantities of motion of the bodies moving in one direction, mi- 

 Mech. 10 



