424 Mr. J. H. Jeans on the 



to specify the radiation, and these vectors, being perpendicular 

 to one another, cannot coincide in direction. 



§ 5. We fall back, therefore, upon the second alternative, 

 and inquire what is the condition that the bands shall be so 

 narrow as to be indistinguishable from the lines. The con- 

 dition is obviously that the mean value of w shall be very 

 small in comparison with p. 



Now in the case of a molecule of a gas, the mean energy 

 of rotation about a principal axis which is not an axis of 

 symmetry is equal to one-third of the mean energy of trans- 

 lation. If &!, w { are the radius of gyration and the rotation 

 about such an axis, 



where c is the velocity of translation, and mean values are 

 taken on each side of the equation. We therefore find that 

 the mean value of w is of the order of c/k. If, in c.G.s. 

 units, we take c = 5x 10 4 , and 2/e = 10 -9 , we get w = 10 u . 



The value of p is of the order of 3 x 10 15 , so that in this- 

 case p and w are of the same order of magnitude, and the 

 bands of the spectrum cannot be so narrow as to be indis- 

 tinguishable from lines. This shows that the vibrator, at any 

 rate in the case of a gas emitting a line-spectrum, is not 

 identical with the molecule. 



§ 6. An exception, however, occurs in the case of mon- 

 atomic gases, and this seems to supply the key to the situation. 

 In a monatomic gas the energy of rotation is known to be 

 very small in comparison with the energy of translation, so 

 that, in our notation, w is small compared with p. We 

 should therefore expect a monatomic gas to emit a pure line- 

 spectrum. 



In general, the condition that to shall be small in comparison 

 with p is, that the rotating body shall be so nearly spherical 

 in shape that the rotation never attains to a value comparable 

 with that which the Boltzmann law of the partition of energy 

 would assign to it. In this case, the energy of rotation is 

 proportional to what has been, in a former paper *, designated 

 as a " subsidiary " temperature. This temperature is given 

 by an equation of the formf 



T = fi/(T), 



where t is the subsidiary temperature, proportional to the 

 mean value of iv*, p is the density of the gas, and /(T) is 

 a function of the temperature of the gas, of which the value 

 is very small. 



* "Distribution of Molecular Energy," Phil. Trans, cxcvi. p. 397. 

 \ I. o. §30. 



