288 Motion of a System of Bodies. 
érdinates of the centre of gravity of the solid; then S(2’?+-4/2)dm 
=S(2?+,y* )dm42XS «dm+2YS,ydm+(X?+Y?)Sdm, but Sdm 
=m, and by the nature of the centre of gravity S,cdm=0, S,ydm=0, 
hence S(x'* +y/*)dm=S(,x?+-,y? )dm+ (X?+Y*)m, (z). 
(z) will enable us to find the moment of inertia relative to an axis 
drawn through any given point when the moment of inertia is known 
for a parallel axis passing through the centre of gravity of the solid; 
it is also evident that the absolute minimum moment of inertia be- 
longs to one of the principal axes which passes through the centre 
of gravity : see Mec. Cel. Vol. I. pp. 75, 76; etc. 
_ It has been supposed in II. that the system revolves about a centre 
of force situated at the origin of the codrdinates, but this is not ne- 
cessary except for simplicity, for the origin may be taken at any 
point, (at pleasure,) provided all the forces are considered as disturb- 
ing forces. 
_ Hence (11.) have place as ies (there being now gant: no 
centre of force at the origin,) and the invariable plane is found in the 
same manner ; the areas 5» ” &c. being now rectilineal triangles in- 
stead of curvilineal sectors, but this does not affect the determination 
of the invariable plane. From what has been said, it is manifest that 
when the system is affected by no foreign forces, there will be a par- 
ticular invariable plane for each point of space. Again (7) are easi- 
tu d?z —zd? 
ly changed to (= a) =Sm(Qzr—Py), Sm(™ a Se 
yd? z—zd?y 
dt? 
=Sm(Rz—Pz), Sm )= Sm(Ry — Qz), (18), where P, 
Q, R, P’, &c. are supposed to phe the same as in (1), (2), (3) given 
r°dv «dy—ydx 
"ssito gt 
at p.40; for as at p- 42, c= » &c. and by resolving 
Q and P at right angles to r, we have the resultant of all the forces 
which affect a unit of m when resolved at right angles to the extrem- 
Qr—Py 
r 
ity of r= =T, .«.Tr=Qz— Py, and in a similar way T’7’= 
—P’y’, and so.on; .*. by substituting these values of c, Tr, ¢’s 
&c. in the first of (7) it will be changed to the first of (18), and in a 
similar manner may the second and third of (18) be obtained from 
the second and third of (7). (18) can easily be found directly from 
(1), (2), (3) ; for multiply the first of (2) and (1) by x and —y, re- 
¢ 
