GENERAL THEORY OF MOMENTS OF INERTIA 303 



Since / is positive for all values of I, m, n, it follows that r 2 is 

 positive for all directions of the radius vector. Thus the conicoid 

 is seen to be an ellipsoid. 



This ellipsoid is called the ellipsoid of inertia of the point 0. 



Equation (125) may be written 



and now expresses that the moment of inertia about any axis 

 through is inversely proportional to the square of the parallel 

 radius vector of the ellipsoid of inertia. 



Principal Axes of Inertia 



248. This physical property of the ellipsoid shows that the 

 ellipsoid itself remains the same, no matter what axes of coor- 

 dinates are chosen. The ellipsoid has three principal axes, which 

 are mutually at right angles. The directions of these axes are called 

 the principal axes of inertia, at the point 0. 



If the principal axes of inertia at the point are taken as axes 

 of coordinates, then the coefficients of yz, zx, xy in the equation of 

 the ellipsoid must disappear. Thus we must have 



D = E = F = 0. 



Taking the principal axes of inertia at as axes of coordinates, 

 equation (124) assumes the form 



The kinetic energy of a rotation of angular velocity II is 



(126) 



where a> v a> 2 , o> 3 are the components of H (see 232). 



