806 Mr. C. G. Darwin on the Reflexion of 



the translations is the volume occupied by the molecule and 

 this is 1/N, where N is the number of molecules in a cubic 

 centimetre. If an incident beam with amplitude Ae ikct falls 

 on the molecule, let the amplitude of the scattered beam at 

 distance r be fAe lk{ - ct ~ r) lr. Here /is of the dimensions of a 

 length, and, apart from the fact that it applies to molecules 

 instead of atoms, is the same as Bragg's (e 2 /mc 2 ) F. It will 

 depend on the orientation of the incident and scattered rays 

 and on the wave-length. It will also vary from one molecule 

 to nnother, according to the chance positions of the electrons, 

 both on account of heat vibration and of the internal motions 

 of: the atoms. 



Let the incident beam come from an anti-cathode at dis- 

 tance H, and let its amplitude at the crystal be A and its 

 glancing angle of incidence be f. Consider the beam 

 scattered to a direction with glancing angle % and azi- 

 muthal angle \\r measured about Oz. Then the source is 

 at — Rcosf, 0, — II sin f, and the point of observation at 

 r cos x cos i|r, r cos % sin ty, — rsin^. The position of a 

 molecule is 



x — aa x + Bb x + JCx "J 



y^aay+fiby+'YtyV, .... (4*1) 



z = a.a 



where a, /3, <y are three integers. For an incident beam 

 j± e ik{ct--R) ^ e wave scattered by this molecule is 



A(/ a ^/r)exp^(^-R aj3y —r aj8y ). . . (4-2) 



Expanding R a/3y and r a/3> , the amplitude of the total scattered 

 wave is 



(A/r)^ /l '^ _E- ^ X / aj 8 7 exp ik[x (cos % cos ijr — cos f) 

 *Py 



+ y cos x sin ^ — z (sin % + sin f )] . 



To find the intensity multiply by the conjugate imaginary. 

 If J = A 2 is the intensity of the incident beam, this gives 



(J/V 2 ) X 2 faPyfa'fi'y' ^P ^ [(^ -^') ( C0S % C0S ^ ~ C0S 

 a/3y a'jS'y' 



where x' = a'a x + /3'b x + ry'c- x , etc. / and its conjugate/' will 

 vary from one molecule to another. We must take the 

 mean of (4*3), allowing for the chance variations of /. 

 This will be done by a double averaging of all possible 



