Motion of a System of Bodies. 45 



2y — y% 



Qx — Py=f — r — )F=0; hence it is evident that 



( . - . . - 



directions of straight lines drawn to the origin of the coordinates ; the - 

 same thing is also evident from the expressions Tr, TV, &c, which ' 

 require F to be resolved in the direction of a straight line which is at 

 right angles to the extremity of r in the plane x,y; .'. resolving F 

 into two, one of which is perpendicular to the plane x, y, and the 

 other in the direction of r; the first of these will not affect T, and 



r 



the second =r xF, but as this acts in the direction of r, it will give 



nothing when resolved in a direction at right angles to r, indeed F 

 will not affect T, since their directions are perpendicular to each 

 other ; hence Tr, TV, &c. are independent of any forces which act 

 on m, m\ &c. in the directions of straight lines drawn to the origin of 

 the coordinates j .\ as before (18) are independent of such forces. 

 Let X, Y, Z denote the coordinates of the centre of gravity of the 

 system, then put #=X-f ,:r, y=Y+ / y, fcsZ-f;, x'—TL+p'* &c. ; 

 by substituting these values of z, y, #, &c. in (18), (since by the na- 

 ture of the centre of gravity Sm/r=0, Sm,y=0, Sm t z=0, Smd 2 i x=0, 



d 2 X df 2 Y d*Z 



&c. also by (4) SmP= jttSw, SmQ= -rz Sot, Sfl»R=-^Sf&;) 



/ %d* u— ,yd 2 x\ 



they will be changed to Smr- -^gy — — ) =Sm(Q/r— P,y), 



Sm \ dt* ) = Sm (R,*- p < z )> Sm [ df* J =Sm ( R # 



Q,*), (19). Hence since (19) are independent of the coordinates 

 of the centre of gravity, the motion of that centre is found in the same 

 way as it would be if all the bodies of the system were united at the 

 centre, and the motion of the system about the centre is found by (19) 

 in the same way that it would be if the centre was at rest, and the 

 same forces were applied, and in the same manner, as when the cen- 

 tre is in motion ; that is, the motion of the system is resolved into 

 two, viz. the motion of the centre of gravity, and the motion of the 

 system about the centre, which are independent of each other. 

 Again, it is evident that (1), (2), (3) will remain the same if the ori- 

 gin of the coordinates has a uniform rectilineal motion in space : 

 .\ (4), and (18) or (7), which are merely transformations of (1), 

 (2), (3) will exist relative to the moveable origin ; supposing the axes 

 of #, y, z to be reckoned from the moveable origin, and each to move 



