406 On the Disj^ersion of Diamond. 



Now, if X2 is a non-vanishing operator (that is to say, not 

 an annihilator), it is, at any rate, self-conjugate and has equal 

 principal values, at least along our previous i, j, k, and 1. 

 But such being the case, Q can only be an ordinary scalar 

 factor, say -sr. Thus 



and since, as on p. 402, (//, 2 - l)E = 9?p, we have, instead of 



2 i $& . ' 



/^ — 1= , the dispersion formula 



yo-7 



A*»-l=- - 9 ^ = — «, . . (14) 



70 — ^-57 — fy Uo — u — war/Vi 



which is precisely o£ the form (13), with the same value 

 of a as in absence of interaction, and with u replaced by 



U =U —-^ (ID) 



Notice in passing that y? — 1 being of the forui ^-- — , 



where A, B are constants, the same is automatically true of 



such refractivity expressions as the much renowned 2 , 9 



of Lorenz-Lorentz. More generally, if tc be any constant 

 number, and if 



9 1 A 



we have also, identically, 



y?-l A 



/A 2 +(*-].) {A+ f cB)-KU , 



that is, again of the form const. : (const. — u). This simple 

 remark may be useful in discussing critically the Lorentz 

 expression (/c = 3) and similar ones appearing in more recent 

 empirical formulae. 



.Returning to formula (15) notice that (yr being at any 

 rate positive) the effect of interaction in such crystals as the 

 diamond would consist in lowering the free frequency (the 

 latter being proportional to the square root of u) or in 

 shifting it towards the red domain of the spectrum. A more 

 detailed discussion of these theoretical relations may be left 

 to the reader. 



February 18th, 1919. 

 "Research Dept., Adam Hilger, Ltd. 



