318 Mr. C. G. Darwin on the 



replaced by the line joining them, Thus the pattern at P is 

 simply the diffraction pattern from a fine line one end of 

 which I x is sharp, while the effect on the other side falls off 

 exponentially. In conformity with our assumption the 

 amplitude of the wave coming from a depth z below I I is to 

 be taken as e —^°osecO £ ^ a ^ coming from J ± (jjl is the 

 absorption coefficient for intensity). The structure that will 

 be observed is thus a band, on one side of which is the 

 diffraction pattern of a straight edge, while in the other 

 direction the intensity falls off exponentially. In any 

 manageable experiment the scale of the diffraction pattern 

 would be only a few seconds, and would be entirely masked 

 by the finite size of the source. For very hard rays the 

 exponential diminution in intensity should be observed. It 

 may easily be seen that at a distance sin cos Oj/x from the 

 sharp edge, the intensity has fallen to 1/e, Thus, for work 

 with bard rays it would be essential to use a very dense 

 crystal, or else to curtail the grating by using a thin one. 

 The line on the photographic plate is, strictly speaking, a 

 conic section and not a straight line, 



It is convenient here to anticipate a future result (§ 6). 

 We shall see that the X rays must be held to have a refrac- 

 tive index which differs from unity by about a millionth. 

 On account of the refraction the position of the line 

 on the photograph is slightly shifted. Let \ 6 be the 

 external wave-length and glancing angle, X\ 0' the internal. 

 Let 1+p be the refractive index. Then \ = {l+p)\' and 

 cos = (1 + p) cos 0' or — 0'= -**p cot 0. The observed 

 position of the line corresponds to ?i\' = 2a sin 0', while that 

 which would be expected is given by n\ = 2asin O , So 

 (1 +p) sin 0' =5 sin O , and so o — 0' —p tan 0. Thus the shift 

 is 



0^-0 o = —p cosec sec 0, 



This result will be proved later ab initio. 



4. Quantitative Method. 



We next consider the case where the reflexion is measured 

 electrically. For this, the information required is quantita- 

 tive. We shall first find the total energy rejected into the 

 electroscope when monochromatic radiation falls on a crystal 

 without any slits. As would be the case in most experiments, 

 we shall suppose the electroscope to be so wide that all the 

 reflected radiation is included ; it is then unnecessary to 

 allow for the fact that the distance of the electroscope is 



