273-] ELLIPSOIDS OF INERTIA. 



of great assistance, similarly as in the problem of finding the 

 centroid (compare Part II., Art. 47). 



Suppose a given mass to have a plane of symmetry ; then 

 taking this plane as the jj/.s-plane, and a perpendicular to it as 

 the axis of x, there must be, for every particle of mass m, whose 

 co-ordinates are x, y, 2, another particle of equal mass m, whose 

 co-ordinates are x, y, z. It follows that the two products of 

 inertia ^mzx and ^mxy both vanish, whatever the position 

 of the other two co-ordinate planes. Hence any perpendicular 

 to the plane of symmetry is a principal axis at its point of in- 

 tersection with this plane. 



If the mass have two planes of symmetry at right angles to 

 each other, then taking one as ^-plane, the other as 2-^-plane, 

 and hence their intersection as axis of x, it is evident that all 

 three products of inertia vanish, 



wherever the origin be taken on the intersection of the two 

 planes. Hence, for any point on this intersection, the principal 

 axes are the line of intersection of the two planes of symmetry, 

 and the two perpendiculars to it, drawn in each plane. 



If there be three planes of symmetry, their point of inter- 

 section is the centroid, and their lines of intersection are the 

 principal axes at the centroid. 



273. Exercises. 



Determine the principal axes and radii at the centroid, the central 

 and fundamental ellipsoids, and show how to find the moment of inertia 

 for any line, in the following Exercises (i), (2), (3). 



( i ) Rectangular parallelepiped, the edges being 2 a, zb, zc. Find 

 also the moments of inertia for the edges and diagonals, and specialize 

 for the cube. 



(2) Ellipsoid of semi-axes a, b, c. Determine also the radius of 

 inertia for a parallel / to the shortest axis passing through the extremity 

 of the longest axis. 



(3) Right circular cone of height h and radius of base a. Find 

 first the principal moments at the vertex ; then transfer to the centroid. 



