Reflection and Refraction of Light. 169 



The last two of these equations give 



L A 2 ^\ ) j. . (14; 



2aC,= | (a-V) ^ AJ-ia-afi ± AJ } v/-l. 



From the first and fourth we get 



A+A^^A^+^gA,', .... (15) 



and from the second and third 

 a" 2 4- V 2 



B U ~tU tv/_ 2 ft// . 



whence, writing as before, 



fi ^a M —a" 



where M is the coefficient of ellipticity, and eliminating B^ B" 

 between the first two of equations (13), 



{a -Mb \^I)A-(a + Mft \^)A y =(«h'-M& V^A/ 



+ (a 2 , -M^V :r i)A 2 ', 

 and from (15) 



2aA = U 1 A 1 ' + U 2 A 2 ', 2aA / = V 1 A 1 ' + V 2 A 2 ', . (16) 



where 



U l ={(aa/ + ^)+M%/- a ) V=T}^| 



*» {ooB^-rO+Main (i-rO V^T} Siiln), 



Xj sin 7^ ? 



V 1 ={(a<-Z> 2 )-M6(a/ + a) ^-1}^=^ 



=^{cos (i + ^J-Msin (t+rj -/-I} S ^^?A> 

 \ sin?'! 



and U 2 V 2 are similar expressions with ( 2 ) written instead 



of (.)- 



First, consider the case in which the incident vibrations 

 are perpendicular to the plane of incidence. 



Then A = 0, and equation (15) and the first of equations 

 (16) give 



Ay= r-72 A x -j- ^2 A 2 , UiAi + U 2 A 2 =0, 



Phil. Mag. S. 5. Vol. 23. No. 141. Feb. 1887. N 



