1892.] and Stability of Dynamical Systems. 353 



certain relations should exist between the 0's, or that some of 

 these quantities should have certain definite values. 



3. As an example of Case III., let us consider the steady 

 motion of an ellipsoid, which is rotating about its centre of inertia 

 under the action of no forces. In this case, there is one ignored 

 coordinate, viz. ■xjr ; and the momentum corresponding to yjr is the 

 constant angular momentum k about OZ. The value of §t is 



* = C cos 2 8 + (A cos 2 cj> + B sin 2 </>) sin 2 6 ^' 



The equations of steady motion are 



d$ =0 - # =0 (0) - 



which are satisfied 



(i) by = 0; 



(ii) by 6 — J?r, and = 0; 



(iii) by 6 — ^tt, and </> = \ir. 



These three conditions respectively correspond to rotation about 

 the least, greatest and mean axes. Hence steady motion is 

 impossible, unless the axis of rotation is a principal axis ; which 

 is a well-known result. 



4. As an example of Case II., we may consider the motion of 

 a solid of revolution or top, spinning about its point. Here <p, as 

 well as y\r, is an ignored coordinate ; the constant momentum 

 corresponding to <j>, is the angular momentum Cco 3 of the top about 

 its polar axis. If k v k 2 be the momenta corresponding to yfr, (f>; 

 the value of 5? will be found to be 



^_ ( /c i- /g 2 cos61 ) 2 , *«' (a\ 



h ~ 2Asin 2 6 + 2G K h 



also V=Mga(l + eos0) (7); 



whence the equation of stead) 7 motion is 



^(ff+TO = (8), 



and will be found to lead to the usual result. 



5. We must consider the condition of stability. 



Let E be the energy in steady motion, and let the suffixes 

 denote the values of the quantities under these circumstances. 

 Then 



