570 Prof. 0. W. Richardson on a Theory of the 



difference between the greatest and the least o£ the half- 

 period areas ; and since, in the example under discussion, the 

 smallest area is negligible compared with the greatest, we 

 may take the greatest of the half-period areas as an upper 

 limit to the value of the integral. Proceeding in this way 

 with the integral remaining after four successive integrations 

 by parts, we find that the value of ifr must lie between 



f = \/^ — x 4-05 x 10 12 [(1--000122) cos /3{r - ri ) 



4- -0091(1 - -000226) sin ^(r^r,) ] 

 and 



ijfr=A A-f& x4-05xl0 12 r(l--000122 + 8-lxl0- 8 )cos/5(r -r 1 ) 



+ -0091(1- -000226 + 2-1 x lO" 7 ) sin /3(r - ri )]. (23) 



It is clear that the only terms which are of any importance 

 are retained if we write 



^= Ay~-|- 4-05 x 10 12 [cos/5(r -r 1 ) + -0091 sin ^(ro-rO]. 



The occurrence of a term containing sin/3(r — r,) shows that 

 the forced vibration is not in complete agreement with the 

 forcing field, but the smallness of the coefficient of 

 sin/3(r — r{) shows that, in this particular instance, the 

 difference of phase is so small as to be of no practical im- 

 portance. In any case the displacement yjr will be capable 

 of being represented with very close approximation by 



^ = ~V^ 3V^7 [(1 +8) cos/3(r<, - ri ) + e **&»-*$ , 



where 8 and e are constants. We have seen that 8 and e 

 w T ere small compared with unity in the typical example just 

 considered. The only important cases where 8 and e will 

 not remain small are when either j3 is nearly equal to <y or 

 the atom B experiences a number of collisions in rapid 

 succession in close proximity to A. There are reasons for 

 believing that these special cases give rise to the broadening 

 of the lines rather than to displacement in any one direction. 

 The point to insist on at present, is rather that these excep- 

 tional cases will not be of such frequent occurrence as to 

 seriously affect the unidirectional shift produced by the more 

 normal case of forced vibrations either in phase with, or in 

 opposite phase to, those of the emitting atom, according as 

 the natural period of the one is greater or less than that of 



