THE MOMENTAL ELLIPSE 227 



If the axes OP and OQ are so chosen that they are the prin- 

 cipal axes of inertia, then, if P is a maximum, Q must be a minimum, 

 and P - Q will be a maximum. 



For P - Q to be a maximum -^3 = 



aa 



That is, - 2(X - Y) sin 2a - 4Z cos 2a = 



or Z = - i (X - Y) tan 2<x 



2 



Thus, if P and Q are the principal moments of inertia, 



P - Q = (X - Y) cos 2<x + (X - Y) sin 2<x tan 2<x 

 (P - Q) cos 2<x = (X - Y)(cos 2 2a + sin 2 2a) 



= (X-Y) 



Then X - Y = (P - Q) cos 2a 

 Also XfY = P+Q 



= P cos 2 a + Q sin 2 a 

 Also Y = 



= P sin 2 a + Q cos 2 a 



Therefore if P and Q are the greatest and least moments of 

 inertia respectively, taken about a pair of rectangular axes which 

 pass through the centroid, the moment of inertia I about any 

 other axis passing through the centroid is given by the relation 

 I = P cos 2 a + Q sin 2 a, where a is the angle between that axis 

 and the axis of greatest moment of inertia. 



123. The Momental Ellipse. Let OX and OY be the principal 

 axes of a plane figure and P and Q be the principal moments of 

 inertia, P being greater than Q. Let OR be any axis making an 

 angle 6 with OX. If I is the moment of inertia of the figure about 

 OR, 



Then I = P cos 2 6 + Q sin 2 6 



Let the lengths OP, OR, and OQ be measured, to the same 

 scale, along the axes OX, OR, and OY respectively, such that 



OR = r 



and OQ = q = ^ ^ 



A being the area of the figure. 



