71] ELLIPSOID OF INERTIA. 



so that the sum of any two of the moments A, B, C is greater than 

 the third. 



If we lay off on the axis a length p and call the coordinates of 

 the point P so determined |, y, g we have 



If we now make the length of OP vary in such a manner that 

 Q 2 Q = 1, we obtain for the coordinates of P the equation 



69) *"(!, ,, g) -4'{ + JBV + <?' 2 + 2.0^ + 2.^ + 27^ = 1, 

 or P lies on a central quadric surface. Since p = = is always real, 



V V 



this is an ellipsoid. It possesses the property that the moment Q 

 with respect to any plane through its center is inversely proportional 

 to the square of a radius vector perpendicular to it. It will be 

 termed the fundamental ellipsoid of inertia at the point 0. It was 

 discovered by Binet. 



In a similar manner the moments of inertia about the various 

 axes are inversely proportional to the square of the radii vectores in 

 their direction of another ellipsoid 



70) F(t, v, Q = A? + Btf + C? - 2Drt - 2E& - 2D^ = 1. 



This is known as Poinsot's ellipsoid of inertia at the point 0. 



Since a central quadric always has three principal axes perpen- 

 dicular to each other (see Note IV), we find that there are at any 

 point in a body three mutually perpendicular directions, namely those 

 of the axes of the two ellipsoids of inertia, characterized by the 

 property that for them the products of inertia D, E, F, are equal to 

 zero. These are termed the principal axes of inertia of the body at 

 the point in question. They have, as shown in 69, the property 

 that if the body be rotating about one of them the centrifugal couple 

 vanishes, so that if the center of mass lies on the axis the body 

 remains rotating about the same axis, unless acted on by external 

 forces. 



The moments A, B, C about these axes are called principal 

 moments of inertia. 



It is important to notice that as we pass along a line which is 

 a principal axis at one of its points, the directions of the axes 

 of the ellipsoids at successive points are not the same, so that in 

 general a line is a principal axis of inertia at only one of its points. 

 We are thus led to study the relative directions of the principal 

 axes at different points of the body. 



