ti 2 T 2 \ 



634 Prof. 0. W. Richardson on Distribution of the 



uniform temperature T is given by eliminating x from the 

 equations 



v=l^{r4jj;&}- • • • (2) 



and 



2 /J^ N\tlOU r x 



In these expressions N is the number of molecules in 

 one gram-molecule (Avogadro's constant), h is Planck's 

 constant, V is the volume occupied by one gram-molecule 

 of the gas, and M is its molecular weight. Assuming that 

 the relation between the pressure p and the heat-energy 

 is invariably given by 



? = 3V' (4) 



at high temperatures when x is small 



P V = NRT, (5) 



and at low temperatures when x is large 



p =aY-* + bVT 4 , (6) 



where 



15/9N\#N 2 P /tn 



a =im^) M" • • (o 



and , 1 16 E/8tt 2 RM\ 3 /q . 



h -5'si'n\~m¥-) (8) 



At high temperatures U is evidently equal to the kinetic 

 energy of the. molecules. At low temperatures there is 

 a term in the pressure which is independent of the tem- 

 perature. 



These relations have been deduced by Keesom on the 

 assumptions : — (1) That the heat-energy of the gas can 

 be analysed into the vibrations in its elastic spectrum, and 

 that the entropy of this system of vibrations can be cal- 

 culated according to the method given by Planck ; (2) that 

 the elastic spectrum is limited by the number of molecules 

 according to the principles successfully used by Debye in 

 calculating the specific heats of solids ; (3) that Planck's 

 hypothesis of zero-point energy has to be taken into con- 

 sideration ; (4) that the interchange of energy between gas 

 and radiation takes place by quanta and that the corre- 

 sponding frequencies are twice as great in the gas as in the 

 radiation in accordance with the principle that the pressure 



