678 Mr. C. G. Darwin on the 



contributes a component in the direction of the transmitted 

 wave. It was this secondary reflexion that was neglected in 

 the earlier work, on the assumption that the radiation 

 scattered by one atom had no effect on the others. The 

 recombination into a single wave o£ the wavelets from the 

 atoms in one plane will not be very complete in the short 

 distance between adjacent planes, but the error in assuming 

 it complete will not be systematic. Moreover, the mutual 

 influence of two planes alone is very small ; it is only the 

 cumulative effect that is important. We thus have a problem 

 very similar to that of the Fabry-Perrot etalon, only with an 

 infinite number of parallel equidistant plates. 



We shall suppose that independently of the scattering of the 

 atoms there is also a small absorption. Thus if a plane wave 

 g ik(ct-x cos e+z sine) f a ]j s on a sm al e plane of atoms, the reflected 

 wave is — iq gfflPt-xeaao-zm*Q anc [ the transmitted wave is 

 (1 — h—iq) eiHGt-xcoao+zsme) m The term Jt re p resen ts the 

 absorption and may be taken as ^ fxa cosec 0. 



We consider a crystal composed of atoms of a single 

 substance arranged in planes at distance a. Let T r represent 

 in amplitude and phase the total transmitted wave just above 

 the (r+l)th plane, S r the total reflected wave in the same 

 position. Then T is the incident wave, and S the reflected 

 wave. S r is derived from two components, the part of T,. 

 reflected by the ?' + lth plane and the part of S,. +1 trans- 

 mitted through it. The latter must be multiplied by a phase 

 factor e ~ ikasine to give its value just below the r+lth plane 

 instead of just above the r + 2th. Thus 



S r =-^T r +(l-A-^ ) e - ika ^ e S r+1 . 



Again T r+ i is made up of the part of T r transmitted through 

 the ?'+ 1th plane and the part of S r+i reflected by it. Putting 

 in the proper phase factors we have 



T r+1 e ika9ine ={l-h-iq ) T r -iqe- ikaa ^ e S r+ i. 



If we eliminate the S's from these difference equations we 

 obtain 



(l-7i-igo)<r f * a,in * (T r _! + T r+1 ) = [l + 5 2 *-»■■«■• 



+ (1 - h - iqof e~ 2ika 8in e ] T,-, 



and the solution is given by T r = T ,i' 5 ", where % is the root of 



(1 — 7t — z^o) e- i7casine lx+ -\ 



= 1 + Q 2 e z 2ikasi » ? + (1- h-igoY g-aaarfn^ . m (2) 



