1893.] On Operators in Physical Mathematics. 105 



These groups are 



I. Methane. 



IT. The three methyl compounds. 



III. Ethane and its derivatives. 



IV. Propane and its derivatives. 



If the members of a group have the same ratio of the specific heats, 

 we know, from a well-known equation in the kinetic theory of gases, 

 that the ratio of the internal energy absorbed by the molecule to the 

 total energy absorbed, per degree rise of temperature, is the same for 

 all. Hence we have the result that, with the single exception of 

 marsh gas, the compounds with similar formulae have the same 

 energy-absorbing power, a result which supplies a link of a kind 

 much needed to connect the graphic formula of a gas with the 

 dynamical properties of its molecules. 



From the conclusion we have reached, it follows with a high 

 degree of probability that the atoms which can be interchanged with- 

 out effect on the ratio of the specific heats have themselves the same 

 energy-absorbing power, their mass and other special peculiarities 

 being of no consequence. Further, the anomalous behaviour of 

 methane confirms what was clear from previous determinations, 

 namely, that the number of atoms in the molecule is not in itself 

 sufficient to fix the distribution of energy, and suggests that perhaps 

 the configuration is the sole determining cause. 



If this is so, it follows that ethane and propane have the same con- 

 figuration as their monohalogen derivatives, but that methane differs 

 from the methyl compounds, a conclusion that in no way conflicts 

 with the symmetry of the graphic formulae of methane and its 

 derivatives, for this is a symmetry of reactions, not of form. 



VIII. " On Operators in. Physical Mathematics. Part II." By 

 OLIVER HEAVISIDE, F.R.S. Received June 8, 1893. 



Algebraical Harmonization of the Forms of the Fundamental Bessel 

 Function in Ascending and Descending Series by means of the 

 Generalized Exponential. 



27. As promised in 22, Part I (' Roy. Soc. Proc.,' vol. 52, p. 504), 

 I will now first show how the formulae for the Fourier-Bessel function 

 in rising and descending powers of the variable may be algebraically 

 harmonized, without analytical operations. The algebraical conver- 

 sion is to be effected by means of the generalized exponential 

 theorem, 20. It was, indeed, used in 22 to generalize the ascend- 

 ing form of the function in question; but that use was analytical. 

 At present it is to be algebraical only. Thus, let 



