72 Statics. 



85 must be zero. Now if we conceive a plane parallel to the direc- 

 tions of the bodies ra, w, o, and passing through G, the moments 

 . with respect to this plane cannot but be zero during the whole 

 motion, for the bodies in their motion are supposed not to alter 

 their distances from this plane ; their distances are therefore 

 constantly the same, and consequently these moments are also 

 constantly the same ; but at the commencement of the motion, 

 that is, when the centre of gravity is in G, the sum of the 

 moments is zero ; accordingly, it is still zero in whatever part 

 of their directions the bodies are ; the centre of gravity is con- 

 sequently in a plane parallel to the directions of the bodies, and 

 passing through the first situation G of this centre. And, as in 

 the reasoning here used, the position of this plane is not other- 

 wise determinate than that it must be parallel to the directions 

 of the bodies, m, n, o, and pass through the point G ; it may be 

 shown, in like manner, that this centre is in any other plane par- 

 allel to the directions of the bodies and passing through the point 

 G ; it is consequently in the common intersection of these planes ; 

 therefore the centre of grayity moves according to GG", parallel 

 to the directions of the supposed bodies. 



(2). The centre of gravity moves uniformly ; that is, if when 

 the bodies m, 71, o, &c., have arrived at A', B', C', &c., we sup- 

 pose that the centre of gravity is in G', we shall have 



GG" : GG 7 : : AA" : AA' : : BB" : BB' :: &c. ; 

 in other words, the spaces described in the same time by the 

 centre of gravity and the several given bodies will be as their 

 velocities respectively. 



Indeed, if we conceive a plane represented by ZX to which 

 the directions of the several motions are perpendicular ; we shall 

 76. have, by the nature of the centre of gravity, 



m X HA + n X IB + o x LC = (m + n -f- o) X KG ; 

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

 m X HA" -f n X IB" -f o X LC" = (m + n + o) X KG". 

 If from the second of these equations, we subtract the first, bear- 

 ing in mind that HA" HA- AA", IB" IB = BB", &c., 

 we shall have 



m X AA" 4- n X BB" o X CC" = (m -f n -f o) X GG"; 



