560 Sir W. Thomson on Periodic Motion. 



d'une trajectoire fermee representant une solution periodique, 

 soit stable, soit instable, il passe nne infinite d'autres trajectoires 

 fermees. Cela ne suffit pas, en toute rigueur, pour conclure que 

 toute region de l'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 connexion with 

 Maxwell's fundamental supposition quoted in § 10 of my 

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

 motion, will, sooner or later, pass through every phase which 

 is consistent with the equation of energy;"! an assumption 

 which Maxwell gives not as a conclusion, but as a proposition 

 which " we may with considerable confidence assert, . . . except 

 for particular forms of the surface of the fixed obstacle." It 

 will be seen that Poincare's " hypothesis, 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 necessary 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 o? distinctes. JSTous les 

 appellerons les coefficients de stabilite de la solution periodique 

 consideree. 



" Si ces n coefficients sont tous reels et negatifs, la solution 

 periodique sera stable, car les quantites fjj et ?)i, resteront 

 inferieures a une limite donnee. 



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

 sens absolu. En effet, nous avons neglige les carres des f et 

 des 7] et rien ne prouve qu'en tenant compte de ces carres, le 

 resultat ne serait pas change. Mais nous pouvons dire au 

 moins que les £ et 97, s'ils sont originairement tres petits, 

 resteront tres petits pendant tres longtemps. Nous pouvons 

 exprimer ce fait en disant que la solution periodique jouit, 

 sinonde la stabilite sSculaire, du moins de la stabilite temporaire." 

 Here the conclusion of § 31 of my present paper is perfectly 

 anticipated and is expressed in a most interesting manner. 

 M. Poincare's investigation and mine are as different as two 

 investigations of the same subject could well be, and it is very 

 satisfactory to find perfect agreement in conclusions. 



* The "trajectoire fermee" of M. Poincare* is what I called a " funda- 

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

 t " Scientific Papers,'' vol. ii. p. 714. 



