Theory of X- Ray Reflexion. 679 



The product of the roots of this equation is unity and that 

 one is to be taken which makes | x \ -< 1. Otherwise the 

 intensity would increase with r. If we substitute back with 

 this solution we find 



°l-x e- ika *™ e (l-h-iq Q y 

 and in particular 



S — iq 



r x- lCe - ika ^ e (i—h-iq y 



(?) 



This expression holds for any angle of incidence. 



We shall now approximate by allowing for the fact that 

 <7, q , h are small and by supposing that the incident wave 

 is very nearly at the angle of best reflexion. Then 6 is very 

 near <£, where ka sin <£ = ?27r — q . The presence here of q 

 represents the shift in the angle of best reflexion due to the 

 refractive index, as explained in the former paper*. We 

 have then 



ka sin 6 = nir — q -f v, 



where v = ka coscf)(0 — <£) (4) 



To the degree of approximation needful we have 

 e- ika » in0 =(-y(l + ig o -iv), 



so that S _ — iq 



%~ l—{-)»x(l-h-iv~) 

 and x is that root of 



(-y{l-h-iv)(x+±\ = l + q* + (l-h--ivy, 



for which | a | *< 1. 



The roots of this equation are very nearly ( — 1)", so to 

 solve it we put £• = (■— ) n (l-— e). 



Substituting in the equation we have 



(l_/^u,)(2 + € 2 ) = l + ? 2 + (l-A~^) 2 , 



so that €= x /{q 2 + (7i + iv) 2 }. 



The ambiguity is to be determined so that the real part is 

 positive. Thus 



h -zk (5) 



* Loc. cit. p. 318. 



