Refractivity and Atomic Interaction. 105 



Therefore, as long as the molecule is not, to all purposes, 

 thoroughly dissociated, neither ayJSlR 3 , o) 2 jdlR d , nor the 

 product of these ratios, appearing in the denominator of 

 (14 a), can be neglected in presence of g^ and w 2 themselves. 

 Notice that, co s obeying the additive law, the contribution 

 due to interatomic cooperation alone can be written, by 

 (Ma), 



_ 4a) 1 6) 2 l + (>i + ft> 2 )/9?R 3 . 



Wa Ws ~ 9?R 3 ' l-4a) 1 a) 2 /9? 2 E 6 " * (14J 



The expression (14 a) given above has seemed interesting, 

 inasmuch as it exhibits the connexion with (and at the same 

 time the departure from) the traditional law of additivity. 

 But this having once been noted, there is another, mathe- 

 matically equivalent, form of the axial molecular refractivity 

 which seems a great deal more interesting, since it brings 

 into evidence the new natural or free frequencies of the 

 molecule as distinguished from those Ov/Yi* \/*li) belonging 

 to the atoms when uncombined. In fact, returning to the 

 intermediate form, written just before (14 a), notice that 

 the squares of the new natural frequencies, let us say 7', y" , 

 are the roots of the quadratic equation 



D=0. 



Thus we can write ~D = (<y' —<y)(y" — y), and therefore 



„ y Bi(y,-7) + B 2 ( 7i - y) + 4B 1 B 2 /R 3 

 °~ Ji " (7'-7)(y' r -y) ' 



which is, essentially, the required form. 



When this is split into partial fractions we obtain ulti- 

 mately, writing again (14 s) for the sake of completeness, 

 the following expressions for the transversal and the axial 

 refractivity : 



L *7 171 , • • • (ib) 



L7 



B' . B" -1 



Tl . I 



+ 



■7 7 -7Jj 

 ay here the constant coefficients B', B" are determined by 



(7' -7") -B' = B 1 (y'-y 2 ) +B 2 ( 7 '- 7l ) -^. r 



(y'-y").B"mB 1 Q rs -r,") + B 1 (y 1 -y")+^\ 



(17) 



