240 VI. SYSTEMS OF VECTORS. DISTEIBUT. OF MASS. INSTANT. MOTION. 



If & < & there are two points for which the ellipsoids of inertia are 

 spheres, namely where d = + ]/a 2 fr 2 . This is the only case, except 

 the above, where there are spheres. 



If we look for ellipsoids of revolution in the general case when 

 , &, c are unequal, we must distinguish between prolate and oblate 

 ellipsoids of gyration. 



1. Prolate. The two equal radii of gyration are the two smaller 

 \ and \ . For these to be equal, we must have k = p. But as A 

 and ^ are separated by c 2 , if they are equal they must be equal 

 to c 2 . In this case the axis of the ellipsoid and one -sheeted 

 hyperboloid are both zero, and the ellipsoid becomes the elliptical 

 disk with axes }/a 2 c 2 , }/& 2 c 2 , forming part of the XF-plane, 

 and the hyperboloid all the rest of the XT- plane. Points lying on 

 both surfaces lie on the ellipse whose axes are "/a 2 c 2 , "J/fr 2 c 2 , 

 which passes through the four foci of the system lying on the X- 

 and F-axes, and is accordingly called the focal ellipse of the confocal 

 system. (We saw by 92] that if A = fi the two surfaces were not 

 necessarily orthogonal.) All points lying on this ellipse have prolate 

 ellipsoids of gyration, the axes of rotation lying in the plane of the 

 ellipse. 



2. Oblate ellipsoids of gyration. In this case we have 



The Y"-axes of the two hyperboloids now vanish. That of one sheet 

 becomes the part of the XZ- plane lying within the hyperbola 



and that of two sheets the remaining parts. The points common to 

 both are those lying on the hyperbola, whose axes are "j/a 2 & 2 , ~|/ft 2 c 2 

 and which passes through the remaining two foci of the system, and 

 is called the focal hyperbola. The axes of revolution of the ellipsoids 

 of gyration lie in the plane of the hyperbola. 



75. Calculation of Moments of Inertia. In the case of a 

 continuous solid, the sums all become definite integrals, as stated 

 in 68. All the preceding theorems of course are unaltered. If 

 the body is homogeneous all the integrals are proportional to the 

 density. Since the mass is likewise, the radii of gyration are in- 

 dependent of the density. We will therefore put Q = 1. 



