262 PROFESSOR POWELL'S REMARKS ON THE 



values (38.) (39.) may obtain ; but in particular cases, or for particular directions of 

 the ray, the greater number of terms of one sign may compensate for the greater 

 magnitudes of those of the other; or the conditions (36.) (37.) may hold good. 



(14.) We now resume the consideration of the formulas before deduced. 



From (28.) we have 



smo— n^^-2^%{p'smH) ^^^') 



Now if the conditions (36.) obtain, that is, if we have 



2 (q sin 2 ^) = 0, and 2 (jo' sin 2 ^) = ; (42.) 



then (41.) will give 



sin & = ; or Z> = (43.) 



But if the polarization be elliptical, b must have a finite value ; or in this case, con- 

 sequently, the conditions (38.) must obtain ; or the non-evanescence of those terms is 

 essential to the investigation for elliptically polarized light. 

 (15.) If the polarization be circular, we have 



TT 



a = |3, and h z= — cos ^ = 0, sin i = 1 ; 



thus the form (26.) becomes 



r 2 (osin 2 ^)l 



. = j ^^^ ^ I (45.) 



L-2 2(ysin2^). J ^ ^ 



But the condition of the non-evanescence of the terms holds good for the same reason as 

 in the last case (46.) 



(16.) For plane polarized light let iS = 0, then the form (26.) becomes 



= 2 (p sin 2 ^) ; (47.) 



or for plane polarization it is essential to suppose the terms evanescent, or the conditions 

 (36.) (37.) to he fulfilled (48.) 



(17.) For unpolarized light we have 



b = sin i -— cos b = } ; 

 but a and |3 are arbitrary. Thus the formula (26.) becomes 



a 2 (» sin 2^) 

 ' 



} . (49.) 



+ |8 2($'sin2^). 



But since this must be true for all values of a and /3 which are independent of p, q 

 and d, it follows that each term separately must be = ; or for unpolarized light it 

 is essential to suppose the terms evanescent, or the conditions (36.) (37.) to be ful- 

 filled (50.) 



(18.) We here take the axis (x) as the direction of the ray, which consequently may 



