1891.] On 'Inatabilitii of Periodic Motion. 199 



ADDENDUM. 



The subject of periodic motion and its stability has been treated 

 with great power by M. Poincare in a paper, " Sur le Probleme des 

 Trois Corps et les Equations de la Dynamique," for which the prize 

 of His Majesty the King of Sweden was awarded on the 21st January, 

 1889. This paper, which has been published in Mittag-Leffler's fc Acta 

 Mathematica,' 13, 1 and 2 (270 4to pp.), Stockholm, 1890, only 

 became known to me recently through Professor Cayley. I am 

 greatly interested to find in it much that bears upon the subject 

 of my communication of last June to the Royal Society " On some 

 Test Cases for the Maxwell-Boltzmann Doctrine regarding Distribu- 

 tion of Energy ; " particularly in p. 239, the following paragraph : 

 " On peut demontrer que dans le voisinage d'une trajectoire fermee 

 representant une solution periodique, soit stable, soit instable, il 

 passe une infinite d'autres trajectoires fermees. Cela ne suffit pas, en 

 toute rigueur, pour conclure que toute region de 1'espace, si petite 

 qu'elle soit, est traversee par une infinite des trajectoires fermees, 

 mais cela suffit pour donner a cette hypothese un haut caractere de 

 vraisemblance."* This statement is exceedingly interesting in con- 

 nexion with Maxwell's fundamental supposition quoted in 10 of 

 my paper, " that the system, if left to itself in its actual state of motion, 

 Avill, sooner or later, pass through every phase which is consistent 

 with the equation of energy ; "f an assumption which Maxwell gives 

 not as a conclusion, but as a proposition which " we may with con- 

 siderable confidence assert, .... except for particular forms of the 

 surface of the fixed obstacle." It will be seen that Poincare's " hypo- 

 thesis, having a high character of probability," does not go so far as 

 Maxwell's, which asserts that every portion of space is traversed in 

 all directions by every trajectory. The conclusion which I gave in 

 13, as seeming to me quite certain, " that every mode differs 

 infinitely little from being a fundamental mode," is clearly a neces- 

 sary consequence of Maxwell's fundamental supposition ; the truth 

 of which still seems to me highly probable, provided exceptional cases 

 are properly dealt with. 



I also find the following statement, pp. 100 101 : " II y aura done 

 en general n quantites a 2 distinctes. Nous les appellerons les coeffi- 

 cients de stabilite de la solution periodique considered. 



" Si ces n coefficients sont tons reels et negatifs, la solution period- 

 ique sera stable, car les quantites and ?/;, resteront inferieures a 

 une limite donnee. 



"II ne faut pas toutefois entendre ce mot de stabilite au sens 



* The "trajectoire fernu'e" of M. Poincare is \vhat I called a "fundamental 

 mode of rigorously periodic motion," or " an orbit." 

 . f ' Scientific. Papers,' vol. 2, p. 714. 



r 2 



