306 ) 



DYNAMICS OF A RIGID BODY. 



I. Moments of Inertia, Principal Axes. 



1463. If m be the mass of an ellipsoid, of which the density 

 at any point is proportional to the product of the distances of the 

 point from the principal planes, the moment of inertia about one 



of the axes is - ; 26, 2c being the axes of the correspond- 



ing principal section. 



1464. If a straight line be at every point of its course a 

 principal axis of a given rigid body, it must pass through the 

 centre of gravity. 



1465. If A, B, C be the principal moments of inertia at the 

 centre of gravity of a rigid body, a s + b 2 + c 2 + r 2 a principal 

 moment at a point whose co-ordinates referred to the principal 

 axes through the centre of gravity are a, b, c; the equation 

 determining r is 



A- r * B-r* C-r* 

 the mass being unity. 



1466. The locus of the points at which two of the principal 

 moments of inertia of a given rigid body are equal is the focal 

 curves of the ellipsoid of gyration for the centre of gravity. 



1467. The locus of the points at which one of the principal 

 axes of a given rigid body passes through a fixtd point, in one of 

 tin- iniiicipiil planes through the centre of gravity, is a circle. 



1468. The loiius of 'he points at which one of the principal 

 axes of a given ripd In dy is parallel to a given straight line is a 

 rectangular hyperbola. 



