MOMENTS AND CENTROIDS 51 
=ABx MN. Therefore ab" x oh’ x oh = AB x MN? = moment of inertia 
of AB about M. a’b’n’ and a’’b’’n” are funicular polygons, of which the first 
determinesthemomentAB x MN, 
and the second determines the 
moment of this moment, namely, 
(AB x MN) =x MN. The lengths 
al’ and a’’b” must be measured 
with the forcescale,and thelengths 
oh and o’h’ with the linear scale. 
Fig. 49 shows the application 
of the method to the determina- 
tion of the moment of inertia of 
the shaded figure about an axis 
aa’ in the plane of the figure. 
The area is divided into parallel b 
strips, and parallel forces AB, 
BC, CD, DE, and EF are sup- 
posed to act at the centres of 
gravity of these strips, the magni- 
tudes of the forces being pro- 
portional to the areas of the 
strips. The sum of the moments 
of these forces about the given 
axis is equal to a‘/” x’oh, and the Fia. 49. 
-sum of their moments of inertia 
is equal to a"f” x o'h’ x oh. The lengths a‘f and a”f” must be measured 
with the area scale and the lengths of and o’’ with the linear scale. 
68. Moment of Inertia—Theorems.—A knowledge of certain 
theorems, which will now be proved, will be found of great use in solving 
problems on moment of inertia. 
_ Theorem I.—If I, and I, are the moments of inertia of a plane 
figure (Fig. 50) about axes OX and OY in its plane and perpendicular 
to one another, and if I, is the moment of inertia 
of the figure about an axis OZ perpendicular to 
the plane XOY, then I,=I,+I,. 
Consider a small element P of the figure, 
whose distance from OY is «2, whose distance 
from OX is y, and whose distance from O is 7, 
and let a denote the area of this small element. 
Then P=22+%%, ar? =az? + ay’, 
Lar? = Lax? + Lay*, therefore I, =I, + I,. Fia. 50. 
Corollary 1.—If OZ is a fixed axis perpen- 
dicular to the plane of the figure, and if OX and OY are any two 
axes in that plane and perpendicular to one another, then I, +I, being 
equal to I, is constant. 
Corollary 2.—Since I,+I1, is constant, it follows that if I, is a 
maximum, I, is a minimum. 
Theorem II.—Let I=the moment of inertia of a surface EF (Figs. 
51 and 53) or a body HK (Fig. 52) about an axis XX passing through 
its centre of gravity G; I, =the moment of inertia of the surface or body 
about an axis X,X, parallel to XX and at a distance 7 from it; A=area 
, 
