the Theory of Double Refraction. 419 



If A £ 2 + A >j 2 + A ?* = A g\ and if X, Y, Z denote the 

 angles which the direction of displacement makes with the 

 coordinate axes, 



A £ = A g cos X Ayj = A g cos Y A ? = A g cos £, 



-X", Y, 2 being the same as in Fresnel's notation, 

 cos 2 X + cos- Y + cos 4 2=1. 



The simplest case, and I believe that implicitly intended 

 by Fresnel, is when the amplitude A g depends only upon the 

 distance ;• from the origin, then 



- € A r 4 + &c. 



1 .2.3.4 dr 

 neglecting the sums of the odd terms * 



ttl 4> r A r, m v[> r A .r 2 A r, &c. 

 we find equations of the form 



ff|-« a* cos 2 jU + a« cos X #4 



' dr dr dr 



T <.- = ^cosr^l + i^cosr-^ 



\f 



dt 4 d ir dr 



a* = -2- 5* -f $ r + ($r)Af JAr« 

 a 2 = i^|$r + (*r) AfJAr 

 #> = 3£ S -f $ r + (*r) A3/ 2 "I Ar* 



I) 2 = ^-^*^r + (*r) A/ J.Ar 4 

 c* = B-xf Q r + (4, r ) a** \ A r 9 



0* = JL;? j"^ r + (vlrr) As 2 | A r 4 



a 9 , & 2 , c 2 being constant and the same as in Fresnel's notation. 



* Mr. Kelland makes 2 <p (r) I x* = 2 <p (r) fy 2 = 2 <p (r) 5 x*. 



Cambridge Trans., vol. vi. p. 159. 

 But this I apprehend is a different case from that contemplated by Fresnel. 



3 H2 



