Oi> PROFESSOR KELLAND ON THE POLARIZATION OF LIGHT 



Now 0_0' 0_0 /+ _0' = 0_0 / + D suppose 



+ 0'= + 0,_ 0,_ (f)'= fy + 0,_ D 



sin 0' = sin 0, + cos 0, . D 



sin + 0' = sin + 0, cos -t- 0, D 



sin 0' sin 0, 

 sinT^T^O = sin (0 + 0,) + C ' L 



But this quantity has to be multiplied into cos + 0' , which is itself of the first 

 order 



v sin 0' sin 0, 



cos 0T0' =^ - = ~ ^= cos + 0' 

 sin + sin + 0, 



omitting quantities of the second order. 

 Hence, dividing by S1 -L~=J', we get 



" sm0 + / 



a, -i -T-, ^ a w cos sin 0, cos 6 



tan 0' cos + 0' + cot 6 cos (0 + 0,) - -^ = 



sin 0,8m 6* 



which is of the/orm given by Mr M'CULLAGII, and is precisely the same as that 

 obtained by the process in Section III. The value of u too is the same as that 

 given by the formula there employed, which shews that we are correct in assum- 

 ing that u depends only on the diiference between the ray and the wave. 



If it seem difficult to leave the quantity T, as part of the undetermined quan- 

 tity, we may easily get rid of the difficulty. In fact, our only reason for adopting 

 this mode of proceeding was, that, since it is requisite to have some one indeter- 

 minate quantity, we may as well have that quantity a compound one as a simple 

 one, provided it simplifies our operations. Let us therefore combine with our 

 former equations, the equation (VII'.), R, + T,=0. 



Then if we multiply equation (1) by 1+*, and equation (4) by 2, and add 

 them, we have 



/T T>N l^ - J. 2 COS 2 0\ f ,. N . , 2 COS 2 0, 1 rp /) 



(I R) (l + #sm0 + , r ) = { (l + *)sm0 / + . , r/ > Tcosfl 

 ' \ sin / sin 0, ) 



{,, , . , . 2 COS 2 0' 1 T>/ Or 

 (1 + *; sm + - E. \ T' sm U , 

 sin 0' 



or if 1 + s=2 + 2 1 where t is very small, 



(I-R) -_ + ; s in0 =Tcos0(- r + ifsin0 / \ -T' sin & __ 

 \sin0 V \sin0, T / \sin0' 



sn 



,,/COS0. Sin Sin0-^- iy -r . ., , \ 



2 I = T cos 6 { & + -r - * -^t- sin 2 - sm j 0, ) 

 \ cos sin 0, sin (p / ) 



T' ' ft' / cos $*' s ^ n ^ / s ' n "^ ~ a"/h r ~a~ 



\cos0 sin 0' sin0' 



