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XL VI II. On Periodic Motion of a Finite Conservative System. 

 By Sir William Thomson*. 



1. ~T'N a recent communication to the Royal Society! I 

 .I. suggested an extension to stable systems in general of 

 the well-known theory of u fundamental modes'" for systems 

 in which the potential energy is a quadratic function of 

 coordinates and the kinetic energy a quadratic function of 

 velocities, each with constant coefficients. This extension is 

 the subject of the present communication to the British 

 Association. 



2. In its title, " finite " means that the number of freedoms 

 is finite and that the distance between no two points of the 

 system can increase without limit. " Conservative " means 

 that the kinetic energy is always altered by the same difference 

 when the system passes from either to the other of any two 

 configurations, whatever be the amount given to it when the 

 system is projected from any configuration and left to move 

 off undisturbed. By "path of a system" we shall under- 

 stand, in generalized dynamics, the succession of configurations 

 through which the system passes in any actual motion : or 

 the group of single lines constituting the paths traversed by 

 all points of the system. By an " orbit " we shall understand 

 a circuital path ; or a "path " of which every constituent line 

 is a complete circuit, and all moving points are always at 

 corresponding points of their circuits at the same time. 



3. It will be convenient, though not necessary, to occa- 

 sionally use the expression, " potential energy of the system 

 in any configuration." When used at all it shall mean the 

 difference by which the kinetic energy is diminished when the 

 system passes to the configuration considered, from a con- 

 figuration or the configuration such that passage to any other 

 permitted configuration involves diminution of the kinetic 

 energy. By " total energy " of the system in any condition 

 will be meant the sum of its kinetic and potential energies. 



4. Theorem of periodic motion. For every given value, E, 

 of the total energy, there is a fully determinate orbit such 

 that if the system be set in motion along it, at any con- 

 figuration of it, with the given total energy, E, it will 

 circulate periodically in it. 



* Communicated by the Author ; §§ 1 ... 10, and §§ 17 ... 22 having 

 been read before Section A of the British Association at its recent 

 meeting- in Cardiff. 



t Proceedings, June 1891, "On some Test Cases for the Maxwell- 

 Boltzmann Doctrine regarding Distribution of Energy." 



