58 APPLIED MECHANICS 
Let the moments of inertia of the figure about OP and OQ be OP=P 
and OQ=Q respectively, and let the moment of inertia of the figure 
about an axis OA making an angle @ with OP be OA=A. ‘Then by the 
formula proved in the preceding Article, A=P cos? 64+Q sin? 6. If P 
and Q are known, and A be calculated for different values of 6 and the 
results plotted, a curve PAQ, called an inertia curve, for the given figure 
is determined. 
Let a denote the area of the given figure. On OP make OP’=7, 
Ee OQ make 0Q'=r,=./, and of OA make OA = re 
Draw A’N perpendicular to OP. Let ON =a, and A‘N=y. 
Fao? cos? 0+ 2 sin? = ee y” ,therefore™. 4 % =1, 
2 a a 2 2 
r @ a ak al at ae 
and therefore the locus of A’ is an ellipse whose principal axes are P’OP’ 
and Q’OQ’. This ellipse is called the momental ellipse of the given 
figure. It will be noticed that any semi-diameter of the momental ellipse 
of a given figure is the reciprocal of the radius of gyration of the figure 
about that diameter. 
74, Determination of the Principal Axes of Inertia of an Unsym- 
metrical Plane Figure.—There are cases in practice in which it is im- 
portant to know the least moment of inertia, or least radius of gyration, 
of an unsymmetrical plane figure, a common example being that of the 
section of an angle-bar 
used as a strut, and this 
form of figure will be 
used to illustrate this 
problem. Fig. 65 shows 
an L-section 3 inches x 
2 inches x $ inch, made 
up of two rectangles. 
In an actual angle-bar 
section there is a fillet 
at the inside angle, and 
the outer inside corners 
are rounded, and these 
modifications of the 
section shown in Fig. 65 
ean be allowed for if 
necessary. 
Find O the centre 
of gravity of the section 
ane any axes OA and Se 
OB parallel to the sides of the section. Determine A and B, the 
moments of inertia of the section about OA and OB respectively. 
Take another axis, OC inclined to OA, preferably at an angle of 45°, 
Find the moment of inertia C of the section about OC. If D is the 
moment of inertia of the section about an axis OD perpendicular to OC, 
then D=A+B-C. . 
Let OP and OQ be the principahaxes of inertia, and let 0 denote the 
