at a Twin Plane of a Crystal. 253 



dicular to the plane of incidence) be M, and that in plane of 

 incidence be N, we have 



M=/x 1 1 + /x 2 e 2 , (56) 



N= > /( i > 2 + r 8 ){® 1 + © 2 }. . . . (57) 



In like manner, if the vibrations of the emergent reflected 

 wave perpendicular and parallel to the plane of incidence be 

 M', JS T/ , 



M'r^tf' + ^fl", (58) 



N'=^(j*+r>){0'+0"}. . . . (59) 



If we are prepared to push to an extreme our supposition as 

 to the smallness of the doubly refracting power, ®, 6 in these 

 equations may be identified with the corresponding quantities 

 in (54), (55) ; for a retardation of phase in crossing and 

 recrossing the stratum alike for all the wares might be dis- 

 regarded. We shall presently return to this question; but 

 we will in the meantime trace out the consequences which 

 ensue when the double refraction, if not extremely small in 

 itself, is at least so small in relation to the distances through 

 which it acts (the thickness of the stratum), that the relative 

 changes of phase may be neglected. Then 



VfiM^je i+ e 2 } 



_ (P2—Pl)VlV<2 _ N 



pi^-Pi) V ' 



We have now to introduce certain relations derived from 

 (37), (38). By elimination of s, we get 



Br./, 2 + A 6{(A-C> 2 +(D-C)p 2 }-Br(^ 2 + r 2 )=0. . (62) 



If we here disregard the difference between p i and p 2 , we may 

 treat it as a quadratic, by which the two values of //, are 

 determined ; and it follows that 



-frli 2 =p 2 + r 2 (63) 



We might have arrived at this conclusion more quickly from 

 the consideration that in the limit the two directions of dis- 

 placement (r, pi,pi), (r,ji2,p 2 ) in the reflected waves must 

 be perpendicular to one another. 



Again, from the general equation (37) we see that 



Br(/, 1 - y u 2 ) + (p 1 2 - r2 2 )D = ; 



