18 Lord Kelvin on the 



molecule of every one of the ordinary gases must have a 

 vastly greater number of atoms in its constitution than those 

 hitherto reckoned in regular chemical doctrine. Suppose, 

 for example, there are forty-one atoms in the molecule of 

 any particular gas; if the doctrine were true, we should have 

 /= 39. Hence there are 117 vibrational freedoms, so that 

 there might be 117 visible lines in the spectrum of the gas; 



and we have k — 1 = — — = '0083. There is, in fact, no 



JL L\) 



possibility of reconciling the Boltzmann-Maxwell doctrine 

 with the truth regarding the specific heats of gases. 



§ 27. It is, however, not quite possible to rest contented 

 with the mathematical verdict not proven, and the experi- 

 mental verdict not true, in respect to the Boltzmann-Maxwell 

 doctrine. I have always felt that it should be mathemati- 

 cally tested by the consideration of some particular case. 

 Even if the theorem were true, stated as it was somewhat 

 vaguely, and in such general terms that great difficulty has 

 been felt as to what it is really meant to express, it would be 

 very desirable to see even one other simple case, besides that 

 original one of Waterston's, clearly stated and tested by pure 

 mathematics. Ten years ago *, I suggested a number of test- 

 cases, some of which have been courteously considered by 

 Boltzmann ; but no demonstration either of the truth or 

 untruth of the doctrine as applied to any one of them has 

 hitherto been given. A year later, I suggested what seemed 

 to me a decisive test-case disproving the doctrine; but my 

 statement was quickly and justly criticised by Boltzmann 

 and Poincare; and more recently Lord Bayleight has shown 

 very clearly that my simple test-case was quite indecisive. 

 This last article of Rayleigh's has led me to resume the 

 consideration of several classes of dynamical problems, which 

 had occupied me more or less at various times during the last 

 twenty years, each presenting exceedingly interesting features 

 in connection with the double question: Is this a case which 

 admits of the application of the Boltzmann-Maxwell doctrine; 

 and if so, is the doctrine true for it ? 



§ 28. Premising that the mean kinetic energies with which 

 the Boltzmann-Maxwell doctrine is concerned are time- 

 integrals of energies divided by totals of the times, we 

 may conveniently divide the whole class- of problems, with 



* • On some Test Cases for the Maxwell-Boltzmann Doctrine regarding 

 Distribution of Energy.' Proc. Roy. Soc, June 11, 1891. 



f Phil. Mag., vol. xx'xiii. 1892, p. 356. " Remarks on Maxwell's In- 

 vestigation respecting Boltzmann's Theorem." 



