I4 2 KINETICS OF A RIGID BODY. [260. 



The constant /c 2 may be selected arbitrarily ; to preserve the 

 homogeneity of the equation it will be convenient to put it into 

 the form /c 2 = J/e 4 , where e is still arbitrary. 



260. As moments of inertia are essentially positive quantities, 

 the radii vectores of the surface 



(10) 



are all real, and the surface is an ellipsoid. It is called the 

 ellipsoid of inertia, or the momental ellipsoid, of the point O. 

 This point O is the centre ; the axes of the ellipsoid are called 

 the principal axes at the point O ; and the moments of inertia 

 for these axes are called the principal moments of inertia at the 

 point O. Among these there will evidently be the greatest and 

 least of all the moments of the point O, the greatest moment 

 corresponding to the shortest, the least to the longest axis of 

 the ellipsoid. 



It may be observed that, owing to the relations of Art. 256, 

 which show that the sum of any two of the quantities A, B, C 

 is always greater than the third, not every ellipsoid can be 

 regarded as the momental ellipsoid of some mass. An ellipsoid 

 can be a momental ellipsoid only when a triangle can be con- 

 structed of its semi-axes. 



261. If the axes of the ellipsoid (10) be taken as axes of 

 co-ordinates, the equation assumes the form 



where J lt 7 2 , 7 3 are the principal moments at the point O. 



By Art. 259 we have ? = K */f= MJ 'I ; hence I=M*/f?. If, 

 therefore, equation (11) be divided by/:) 2 , the following simple 

 expression is obtained for finding the moment of inertia, /, for 

 a line whose direction cosines referred to the principal axes 

 are , /3, 7, 



"- (12) 



