686 Mr. C. G. Darwin on the 



focussing, since (though for a different reason) his photo- 

 graphic plate and source were at the same distance from the 

 crystal. 



It is quite possible that a given crystal surface should 

 systematically reflect more than its due share of radiation. 

 For example, if it were of a wavy form, each separate wave 

 would reflect a ray. But we can show that on the average 

 there is no improvement in the reflexion when the surface is 

 supposed divided into small jdates, the normals of which 

 deviate from their mean direction in a random manner. In 

 such an indefinite question as the present it is useless to 

 proceed with any great rigour, and we shall be content with 

 a rather general argument. In the first place, there is no 

 need to allow for the fact that the normals of the plates 

 deviate from the plane of incidence of the rays. The only 

 effect of this is to shift the ray to a different part of the line 

 of reflected rays. In considering the reflexion from a set of 

 plates, the normals of which all lie in the plane of incidence, 

 it will be sufficient to take it that a ray is reflected to the full 

 extent given in (1 1) , when a line can be drawn from the sourc e 

 to the plate, making exactly an angle cf>, the reflexion angle, 

 with its plane. For a plate in any fixed position on the 

 crystal there will be a certain small range of directions of 

 the normal such that a line can be drawn from a given 

 source to make angle </> with the plate. This range is limited 

 by the two positions when the line cuts the plate at either of 

 its two edges, and therefore the range of inclinations of the 

 plate which can give a reflexion is proportional to, its breadth. 

 The chance of a reflexion is thus not altered by cutting the 

 plate in half, for if this is done either of the halves must be 

 aimed in the proper direction with just twice the accuracy, 

 that is to say each half is just half as likely to give a re- 

 flexion. Thus there is on the average the same probable 

 number of reflexions when the crystal is broken into many 

 plates as when it is broken into few, or finally as when it 

 is perfect. We conclude that there is no average improve- 

 ment or deterioration of reflexion when the surface of the 

 crystal is broken up. 



When we come to consider the inside of the crystal the 

 matter is quite different. We saw in § 3 that if the crystal 

 is perfect all the radiation that can be reflected, is so, long- 

 before the depth at which rays at a different angle are ap- 

 preciably absorbed. Now if the crystal is twisted internally 

 these unabsorbed rays may come on a part of it at the right 

 angle, and so give rise to a second reflexion. We must 

 estimate how this will affect the matter. Suppose d to be a 



