812 Mr. 0. G. Darwin on the Reflexion of 



according to the error law. Let a be the scatter of the 

 blocks — that is, the departure o£ mean square of the normal 

 of a block from that of the conglomerate. Then for (5*1) 

 we must write 



V exp-(P + m 2 )/2<x 2 



&£ 2tt<7 2 ....... <o <; 



It is now possible, though still tedious, to work out (5*3) 

 down to the last integration, which involves an error function. 

 Approximating for this when u is small we find 



jVQfgg/L w/a* \ . (5 . 8) 



V 27TO- L v2it(t . k sin . f J 



as the expression corresponding to (5*5) ; so the validity of 

 (5*5) depends on neglecting the second term in the bracket. 

 Thus for a first-order reflexion a/f must be small compared 

 to a. In Bragg's experiments a was of the order of 1°; so 

 to get an accuracy of 1 per cent, f/a must be of the order 

 of 10 4 . For spectra of higher orders the conditions are less 

 exacting. 



From the general appearance of the work one may hazard 

 a guess that the approximation will be true over a much 

 wider range of values, and would cover the case of a crystal 

 imperfect by warping. It is, of course, possible always to 

 define a function Gr(w) so as to satisfy (5 5) and it will 



probably be always true that \ G(u)du=Q; but the 



—oo 



important point is that G(V)/Q should depend only on 

 the structure of the conglomerate, for only so will it be 

 possible to pass from reflexion of one order to one of another 

 and from one set of crystal planes to another. 



6. Extinction. 



The calculations have so far dealt with crystals which are 

 so thin that absorption and extinction can be neglected. 

 It is now necessary to inquire to what extent this is 

 justified. In D.ii. a study was made of the reflexion from 

 an infinitely deep perfect crystal, and it was shown that the 

 reflexion is practically perfect when the glancing angle differs 

 from 6 by less than q/hacosO, where <^ ~ N/Aa cosec is 

 the coefficient of reflexion for a single plane (a quantity of 

 zero dimensions) ; while on either side of this band it falls 



