74] DISTRIBUTION OF PRINCIPAL AXES. 239 



The direction cosines are then given, according to 95) and 98), by 



f ( 2 + r*- V) - f (V + r* - V) = f (c 2 + r* - kf), 

 100) S(^ + ^-V) = (6 2 + ^-V) = f(c 2 + - 2 -V), 



that is 



etc. 



Hence the principal axes of inertia at any point are normal 

 to the three surfaces through confocal with the ellipsoid of gyration 

 at the center of mass. This theorem is due to Binet. 



Since A, > /z, > v, the least moment of inertia is about the normal 

 to the ellipsoid, the greatest about the two -sheeted hyperboloid, and 

 the mean about the normal to the one- sheeted hyperboloid. 



We have 



It* + ^ + fa* = 3r 2 - (I + 11 + *,). 



But the sum of the three roots is the negative of the coefficient 

 of 2 in the cubic 83), 



I -f 11 + v = x* + f ' + z 2 - (a 2 + V + c 2 ), 

 101) ^ 2 + ^ 2 2 + ^ 3 2 = 2r 2 + a 2 + 6 2 + c 2 . 



Thus the sum of the principal moments of inertia is the same for 

 all points lying at equal distances from the center of mass. 



It is now easy to see that any given line is a principal axis for 

 only one of its points, unless it passes through the center of mass, 

 when it is such for all of its points. It is also evident that not 

 every line in space can be a principal axis. 



If the central ellipsoid of gyration is a sphere, all the ellipsoids 

 of the confocal system are spheres, and all the hyperboloids cones. 

 Every ellipsoid of inertia is a prolate ellipsoid of revolution, with 

 its axis passing through the center of mass. 



If the central ellipsoid has two equal axes, the ellipsoids of 

 inertia for points on the axis of revolution are also of revolution. 

 If the distance of a point on this line from the center of mass is d> 

 and the moment of inertia about it is M Jc^ 



