818 Mr. C. Gr. Darwin on the Reflexion of 



multiple reflexions in the successive layers. The multi- 

 plicity is of a different type from that of (6*1) because the 

 rays are not now coherent. The problem of these multiple 

 reflexions would be exceedingly difficult if it were treated 

 exactly ; for each layer will, on account of diffraction, 

 spread out incident parallel rays into a certain range of 

 angles and so will continually change the angle at which 

 they attack the successive layers. But, if (as assumed 

 in § 5) the crystal is so imperfect that diffraction does not 

 change the direction of the rays to an extent comparable 

 with the scale of variation in the orientations of the blocks, 

 then it will be legitimate to regard the reflected rays as 

 coming plane parallel off the crystal (at an angle exactly 20 

 to the incident beam). In consequence of this it will be 

 possible to replace a highly complicated system of integral 

 equations by differential equations of a simple type. 



Suppose that a plane incident beam of total power I strikes 

 a deep crystal at angle + u to the face, and let E„ be the 

 total power reflected. Let I u (z) and E„(^) be the powers of 

 the incident and reflected beams at a depth z inside the 

 crystal. Suppose that the area of face they strike is B, and 

 consider the effect of a layer of thickness Sz. The incident 

 beam has cross-section Bsin(# + u) and so its intensity is 

 I„(£)/Bsin (0 + m). The power reflected by the volume ~B8z 

 will, by (6'H), be therefore I u (z)8zG(u) cosec (0 + u). The 

 incident beam will lose the same amount through extinction, 

 while through absorption it will lose I u (z) 8z /ul cosec (0 + u) . 

 In the same way the reflected beam will be partly reflected 

 back again. To treat of it we must regard the conglomerate 

 upside down — that is, for F(W, I, m) in (5"1) we must write 

 F(W, —I, —m) and also for u, —u. Thus Gr(u) will be 

 unaltered in form, and the beam E u (z + 8z), which is coming- 

 outwards through the layer Sz, throws an amount 



E u (z+8z)Sz G(u) cosec (0-u) 



back into the incident direction. Corresponding to this 

 there is an amount 



E u (z + Sz) Sz{fi + G(u)} cosec (0-u) 



absorbed and extinguished. Balancing up the gains and 

 losses we arrive at a pair of equations, 



HIM _ j^+G(ul U£) + G(u) 



~dz sin(0 + u) sin (0— u) 



dE tt (s) _ __ p + Gr{u) F G(u) 



B* ~ sin (0-u) " K, " f "sin(0 + w) mW 



