1900.] on the Dynamical Theory of Heat and Light. 377 



the discrepancies from the values, calculated according to the Boltz- 

 mann-Maxwell doctrine, are real and great ; and that in each case, 

 diatomic, triatomic, and tetratomic, the doctrine gives a value for 

 k — 1 much smaller than the truth. 



§ 26. But, in reality, the Boltzmann-Maxwell doctrine errs enor- 

 mously more than is shown in the preceding table. Spectrum 

 analysis showing vast numbers of lines for each gas makes it certain 

 that the numbers of freedoms of the constituents of each molecule is 

 enormously greater than those which we have been counting, and 

 therefore that unless we attribute vibratile quality to each in- 

 dividual atom, 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, 

 r & 120 



in fact, no 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 experimental verdict not 



true, in respect to the Boltzmann-Maxwell doctrine. I have always 



felt that it should be mathematically 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 Bayleigh f 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 different 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 ? 



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

 tribution of Energy.' Proc. Roy. Soc, June 11, 18U1. 



f Phil. Mag., vol. xxxiii. 1892, p. 356. 'Remarks on Maxwell's Investigation 

 respecting Boltzmann's Theorem.' 



Vol. XVI. (No. 94.) 2 c 



