Professor Mac Cullagh's Lectures on Rotation. 151 



axis of x, the following relation between these two periods may be readily de- 

 duced from (15): 



f _ rp be 



If the axis of moments, and consequently the axis of revolution, be situated near 

 the axis of greatest or least inertia, it will always continue near tliis axis ; if, 

 however, it be situated near the mean axis, tlie movement of the body will be 

 determined by the following construction. Let tlie two cyclic planes of the 

 ellipsoid be drawn through the mean axis ; they will divide the ellipsoid into 

 two regions, in one of which is situated the axis of maximum inertia, and in the 

 other the axis of minimum inertia. The spherical conic described by the axis 

 of principal moments will have the first or second of these axes for its internal 

 axis, according as R is greater or less than the mean axis. If the axis of prin- 

 cipal moments lie in one of the cyclic planes, the spherical conic becomes a 

 circle, and its two projections become identical with itself (fig. 2, p. 148) ; the 

 expressions (14) are reduced to the form 



Kdt= '^'^ • 



COS0 



wliich when integrated gives 



Kt-V A =^ log cot (^ -| 



or. 



»(M) = ».(j-*j)."; (17) 



00 being the value of corresponding to < = 0, and K being expressed by the 

 following quantity: 



ft b^ac 



It is evident from the equation (17), that the axis of moments will coincide with 

 the mean axis of inertia at the end of an infinite time. 



