CHAPTER VI 



MOTION WHEN THERE ARE NO STABLE CONFIGURATIONS 



OF EQUILIBRIUM 



117. The result obtained in the last chapter for the rotational problem 

 combined with those previously obtained in Chapter III for the tidal and 

 double-star problems, has now established that 



In all the three problems under consideration there are no figures of stable 

 equilibrium except ellipsoids and spheroids. 



In each of these problems the succession of states has been determined 

 by the continuous variation of a parameter the angular momentum in 

 the rotational and double-star problems, and the distance R in the tidal 

 problem. And in each case it is quite possible for this parameter to vary 

 to beyond the limits within which stable configurations are possible. We 

 must accordingly try to obtain what information we can as to the changes 

 to be expected after this limit is passed. 



Poincare*, writing with special reference to the rotational problem, re- 

 marks that if the pear-shaped figure proved to be unstable, " la masse fluide 

 devrait se dissoudre par un cataclysme subit." The pear-shaped figure has 

 now been proved to be unstable, and we must examine the nature of the 

 cataclysm. The situation is similar in the two other problems; when the 

 two masses concerned in either approach one another to within less than 

 a certain distance no configurations of stable equilibrium are possible, and 

 a cataclysm occurs. 



The term cataclysm provides a convenient name for the events which 

 take place when stable equilibrium becomes impossible, but we must notice 

 that mathematically nothing more sensational happens than that a statical 

 problem gives place to a dynamical one. A statical problem may or may 

 not admit of solution, but a dynamical problem must always have a solu- 

 tion. Equations of motion which cannot be satisfied with the accelerations 

 put equal to zero, necessarily admit of solution when the acceleration terms 

 are restored. 



We now consider the three problems in turn, beginning with the tidal 

 problem. 



* Letter to Sir G. Darwin, quoted in the latter's Coll. Works, in. p. 315. 



