SPECTRAL LINES 447 



latter. It will be necessary then to calculate the reaction at A 

 due to the forced vibrations set up in the atom at B by a given 

 vibration at A, to sum this up for all the atoms B which 

 occur, and to find the effect of the resultant reaction on the 

 period of A." 



Working on these lines, by a straightforward but rather 

 tedious piece of analysis Richardson arrived at a displacement 

 dX of the wave length X given by 



dX e'AV - i) 

 X ' 67r 2 mc 1! a 3 



where fi is the refractive index of the surrounding gas, c is the 

 velocity of radiation in free aether, e and m denote the electronic 

 charge and mass, and a is the radius of a sphere within which it 

 is impossible for the centre of an atom of class B to lie and is 

 supposed to be between a and 2a, where a is the atomic radius. 



Supposing that the surrounding gas is air at the arc tempera- 

 ture (2730 abs.), so that ji 2 — 1 = 5-9. io -5 , and taking for a a mean 

 value of r5.io~ 8 cms., Richardson calculated that for a wave 

 length X = 4. io -5 cms., dX/X = 9. io -6 , which is about one hundred 

 times as large as the average value obtained experimentally 

 for the wave length used. The cause of this discrepancy is 

 easily found. Richardson first calculated the effect of one atom 

 of class B on the atom A, and then obtains the effect for the 

 whole of the surrounding gas by multiplying this by the number 

 of atoms per unit volume and integrating throughout the whole 

 volume external to the atom A. Thus the surrounding medium 

 was implicitly treated as continuous right up to the vibrating 

 molecule A, which, as has been remarked above, is not per- 

 missible. If, as in Larmor's theory, the surrounding medium be 

 treated as continuous outside a sphere of radius ka concentric 

 with the atom, and if k be calculated from the above formula 

 using the observed mean value of dX/X, it is found that k is 

 approximately 5, which is too small since the atoms of the gas 

 are, on the average, under these conditions at a distance apart 

 which is equal to fifty or sixty times the atomic radius. If, on 

 the other hand, a larger and more probable value of k be used, 

 dX/X is again much smaller than the observed value and the 

 theory cannot be regarded as affording an adequate explanation 

 of the pressure shift. Even apart from the numerical disagree- 

 ment there are the additional objections that, as with the theory 

 of Larmor, it gives a shift proportional to the square of the 



