232 VI. SYSTEMS OF VECTORS. DISTBIBUT. OF MASS. INSTANT. MOTION. 



72. Ellipsoid of Cry ration. The moment of inertia about any 

 axis may be considered equal to that of a particle whose mass is 

 that of the body placed at a distance Js from the axis, such that 

 K = MJc 2 . It is called the radius of gyration for this axis. The 

 radii of gyration about the principal axes of inertia at any point are 

 called the principal radii of gyration for that point. If we call their 

 lengths a,l),c we have 



and 70) becomes 



Another ellipsoid besides Poinsot's, which referred to its axes is 



72) F(x, y, e) = Ax 2 + By 2 + Cz 2 =l 



is sometimes convenient. If at any point x, y } z on Poinsot's ellipsoid 

 we draw the tangent plane, and from the center let fall a perpen- 

 dicular upon it, its length p will be the projection of the radius 

 vector r on a line parallel to the normal, 



73) p = x cos (nx) + y cos (ny) -f z cos (nz). 

 But since 



Ax 



rjA\ 



this gives for the ellipsoid 



dF . dF . dF 



Thus the direction cosines of p are, by 74), 



a' = cos (nx) = Apx = Apr a, 

 75) j8 f = cos (ny) = Spy = Bprfi, 



y' = cos (nz) = Cpz = Cpry. 



If on the perpendicular we mark off a point P' at a distance 



-R 2 

 OP' = r ' = 9 and call its coordinates x',y\z\ we have 



