I4 8 KINETICS OF A RIGID BODY. [270. 



270. The surface (17) has variously been called the ellipsoid 

 of gyration, the ellipsoid of inertia, the reciprocal ellipsoid. We 

 shall adopt the last name. The semi-axes of this ellipsoid are 

 equal to the principal radii of inertia at the point O. The 

 directions of its axes coincide with those of the momental 

 ellipsoid ; but the greatest axis of the former coincides with 

 the least of the latter, and vice versa. 



By comparing the equations (12') and (18) it will be seen that 

 q is the radius of inertia of the line (, /?, 7) on which it lies. 

 Thus, while the radius vector OP = p of the momental ellipsoid is 

 inversely proportional to the radius of inertia, i.e. /o = e 2 /q, the 

 reciprocal ellipsoid gives the radius of inertia <\for a line 1 as the 

 segment cut off on this line by the perpendicular tangent plane. 



271. We are now prepared to determine the moment of 

 inertia for any line in space. Let us construct at the centroid 

 G of the given mass or body both the momental ellipsoid and 

 its polar reciprocal. The former is usually called the central 

 ellipsoid of the body ; the latter we may call the fundamental 

 ellipsoid of the body. As soon as this fundamental ellipsoid 



^ + ^ + i=i 



q? <?<? 3* 



is known, the moment of inertia of the body for any line what- 

 ever can readily be found. For, by Art. 270, the radius of 

 inertia q for any line / passing through, the centroid is equal 

 to the segment OQ cut off on the line / by the perpendicu- 

 lar tangent plane of the fundamental ellipsoid; and for any 

 line / not passing through the centroid the square of the 

 radius of inertia can be determined by first finding the square 

 of the radius of inertia for the parallel centroidal line / and 

 then, by Art. 250, adding to it the square of the distance d 

 of the centroid from the line /. 



272. In the problem of determining the ellipsoids of inertia 

 for a given body at any point, considerations of symmetry are 



