274 CELESTIAL MECHANICS. I. vii. 30. 



with respect to the body, we must have d/>=0, dg=0, and 



jj j^ 



dr=0, whence, from the equations {D) we have — pr, — rq 



(J j^ j[ (J 



=0, -— — rpziO, and — -—.pq^Q: and, in the general 



extent of the theorem. A, B, and C being all unequal, it 

 follows that two of the three quantities p, q, and r must 

 vanish, which supposes the momentary axis of rotation to 

 coincide with one of the principal axes. 



If two of the three quantities. A, B and C, are equal, 

 for example if ^ = ^, these three equations only give us 

 rpzuO and pgrizO, which will be true if jp only be supposed 

 to vanish, so that the axis of rotation may be perpendicular 

 to the third principal axis, and it has been already shown 

 that, in this case (351), all the axes so situated are principal 

 axes. And again, if A, B, and C are all equal, the three 

 equations will be true, whatever may be the values of/>, q, 

 and r ; but in this case all the axes are principal axes (352). 



Hence it follows that the principal axes only can be 

 permanent axes of rotation : but they do not possess this 

 property in the same manner: the rotation round that 

 axis, with regard to which the rotatory inertia is inter- 

 mediate between the two others, may be disturbed in a 

 sensible degree by the slightest cause, so that such a mo- 

 tion is possessed of no stability. 



Stability consists in such a state of a system, that when 

 it is very slightly deranged, the derangement can only re- 

 main extremely slight, and the system will oscillate about 

 the state of stability. Thus if we imagine the momentary 

 axis of rotation to be infinitely little removed from the 

 third principal axis, in this case the values of ^ and r will 

 always remain infinitely small, and the momentary axis will 

 only make excursions of the same order about the third 



