Motion of a System of Bodies. 45 
only, we have Qr:—Py= ine hence it is evident that 
(18) are independent of any forces which act upon m, m’, &c. in the 
directions of straight lines drawn to the origin of the codrdinates; the 
same thing is also evident from the expressions Tr, T’r’, &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 planez, y; -’. resolving F 
into two, one of which is perpendicular to the plane x, y, and the 
other in the direction of 7; the first of these will not affect T, and 
the second =F x F, but as this acts in the direction of r, it will give 
nothing when resolved in a direction at right angles to 7, indeed F 
will not affect T’, since their directions are perpendicular to each 
other ; hence Tr, T’r’, &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; .*. as before (18) are independent of such forces. 
Let X, Y, Z denote the codrdinates of the centre of gravity of the 
system, then put z=X-+,r7, y=Y+y, z=Z4+ 2, c’!=X+,2', &e.; 
by substituting these values of 7, y, z, &c. in (18), (since by the na- 
ture of the centre of gravity Smz=0, Sm,y=0, Sm,z=0, Smd? +=0, 
d?X 2 iy: d?Z, 
&c. also by (4) SmP =, 5m, SmQ=TFs Sn, SmR =e Sm ;) 
i jd? y — yd? x : 
they will be changed to Sm( Ht  *) <sm(Q2—Py), 
Dida ae Bogs Ja stiles 2 
sn( 2 is sel = =Smn(R,r-P,z), Sin (x = Tee S (Ry 
—Q2z), (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 coérdinates 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 x, y, 2 to be reckoned from the moveable origin, and each to move 
