46 Motion of a System of Bodies. 
parallel to itself. If the system is not subjected to the actions of any 
foreign forces, (11) will exist relative to the moveable origin, whether 
the changes of the motions of the bodies are finite in an instant or 
not, and A, ,A, ,,A, will each be invariable during the motion of the 
system; .°. (16) will each be invariable, hence the invariable plane 
always passes through the moveable origin and moves parallel to it- 
self. It has been proved when the system is not subjected to the ac- 
tions of any foreign forces, that the centre of gravity is either at rest 
or moves uniformly forward in a straight line; .°. by fixing the ori- 
gin at the centre of gravity, the invariable plane is either at rest or 
moves parallel to itself. Let the origin be at the centre of gravity, 
and suppose that the system is not oe to the actions of any 
ape dt 
UA; Sn( m =,A, (20). Now we have m'((z’—2)dy'— 
(y'— y)dx') =m'(2'dy' — y'dz')+-m'(yde' — xrdy’),m'((x""—2) dy" —(y" 
— ye") =m" (x''dy" —y"da") +m" (yde" —ady"), and so on for all - 
the bodies except m: hence Sm’( (a — x) dy’ — (y' — y) dz’) = 
Sm’ ( x’dy' — y'dx') + Sm’ ( yda' — axdy’), but Sm'yda' = ySm'da’, 
Sm'axdy’=a«Smdy', and by the nature of the centre of gravity Sm'da’ — 
foreign forces, then by (11) Sm “oS 
—«x)dy'— dz’ 
= —mdx, Sm'dy' = — mdy; hence eon (a B) LS mie -} 
—x')dy —(y—y!)dee\ 
=mA, in the same way n'Sn(C— VW) —m'A,and 
(A al v di foiaher Giles: dix’ 
so on for all the bodies; .°. mon (DV i (yi 9) ee 4+ 
on) + &. =mA + mA + &e. a 
ca ce) =) — asm, (21), in the 
—«).(dz'—d —z).(da’ —d. 
same way Smm’ oe a: es ceded =) = ,ASm, 
dz’ —z).(dy'—d 
an) (ee mee t2) 5 MO 2).(dy' = D) Lash, (22). 
Let the plane x, y be a invariable plane, then A is a maximum, 
and ,A, ,A are each =0; .°. the first member of (21) is a maximum, 
and those of (22) are Sach =0. Leta plane be drawn through any 
body of the system parallel to the plane 2, y, then the first member 
of (21) is a maximum relative to the parallel plane ; for its value is 
