680 Mr. C. G. Darwin on the 



The earlier paper was written under an assumption which 

 may be seen to be equivalent to taking q much smaller than 



h, so that y~ = t y-r7 — ^r*. It is quite possible to evaluate 



the expressions required with any values of h and q, but the 

 formula? involve elliptic functions, so that their numerical 

 values are not easy to see. Now as we saw q is probably 

 about 2xl0" 4 while h is only 4 x 10" 6 , so that not much 

 error is introduced by supposing hjq negligible. It would 

 not, however, have been permissible to have supposed that 

 h vanished at the beginning of the work, because if this were 

 done it would be found that for some angles the reflexion 

 tends to no definite value as the number of planes tends to 

 infinity. 



In discussing the ambiguity of \/{q 2 + (h-\-iv) 2 } when h 

 vanishes it will be convenient to suppose q positive. We do 

 not know whether this is true, but if q is really negative the 

 modification is very simple. When —q<v*<q we have 

 simply \/(q 2 — v 2 ), the positive square root being taken. 

 When v =>■ q we must write the expression in the form 

 ±iy/(v 2 —q 2 —2ivJi), and if the radicle is expanded it will be 

 seen that the proper value is + i s/ (v 2 — q 2 ) . Without the 

 presence of h this could not have been determined. Similarly 

 when v<: — q we have to take — i^/(v 2 — q 2 ). Thus the 

 amplitude of reflexion is 



a 



lor v -< —a 



v-i^/i^-v 2 ) 

 v+^/{v 2 -q 2 ) 



for —q<v<:q \> . . . (6) 

 for q < v 



To express the intensity of reflexion we take the moduli 

 of the squares of these quantities. In the middle region 

 this is unity and reflexion is perfect. Now v = ha cos cf){0 — <p), 

 so this is the region 



= (f> + s, where s = q/ka cos <p. 



If we take the reflexion in the first order of rocksalt for the 

 platinum radiation (3 (X=l*ll x 10~ 8 cm.) we find s = '6". 

 For the second order it is about half this. 



On account of the perfect reflexion the transmitted wave 



* This is equivalent to one of the equations on p. 322 of the earlier 

 paper. 



