SI Sir W. Thomson. On the Maxwell-Boltzmann [June 11, 



whole two hundred million million globules will be certainly small in 

 comparison with the ultimate average kinetic energy of the single 

 shell. It may be because each globule of 6 is free to wander that 

 the energy is lost from the shell in that case, and distributed among 

 them. There is nothing vague in their motion allowing them to 

 take more and more energy, now when they are connected by the 

 massless springs. Tf we suppose the motions infinitesimal, or if, 

 whatever their ranges may be, all forces are in simple proportion 

 to displacements, the elementary dynamical theorem of fundamental 

 modes shows how to find determinately each of the 600 million million 

 and six simple harmonic vibrations of which the motion resulting from 

 the prescribed initial circumstances is constituted. It tells us that the 

 sum of the potential and kinetic energies of each mode remains 

 always of constant value, and that the time-average of the changing 

 kinetic energy during its period is half of this constant value. 

 Without fully solving the problem for the 600 million million and 

 six co-ordinates, it is easy to see that the gravest fundamental mode 

 of the motion actually produced in the prescribed circumstances 

 differs but little in period and energy from the single simple harmonic 

 vibration which the two shells would take if the globules were 

 rigidly connected to them, or were removed from within them, and 

 the other initial circumstances were those of 6. But this conclu- 

 sion depends on the forces being rigorously in simple proportion to 

 displacements. 



10.* In no real case could they be so, and if there is any deviation 

 from the simple proportionality of force to displacement, the inde- 

 pendent superposition of motions does not hold good. We have still a 

 theorem of fundamental modes, although, so far as I know, this 

 theory has not yet been investigated. f For any stable system moving 

 with a given sum, E, of potential and kinetic energies, there must in 

 general be at least as many fundamental modes of rigorously periodic 

 motion as there are freedoms (or independent variables). But the 

 configuration of each fundamental mode is now not generally similar 



* Sections 10 to 17 added July 10, 1891. 



' f It is similar for adynamic cases, that is to say, cases in which there is no 

 potential energy, as, for example, a particle constrained to remain on a surface and 

 moving along a geodetic line under the influence of no " applied " force. 



