488 Royal Society : — 



Ficsnel's formulae (2). Put 



i — i'=d, i-\-i =<r ; 



then, from (1), 



dl (W dd drr . ., , 



r :=t r.=; : — r 7=; r-rr: .; = COS l COS l' da), Suppose, 



tan i tan r tan i — tan z' tanz + tan-*' 1A 



whence 



^ = sin0^w, <7<r = sin <n/w ; .... (22) 



and we see that i and w increase together from i~0 to t=— . We 

 have also 



n^!^! , rf o= 2shl0 (sin er cos 0^0 -sin cos trda) ==^!i(cos 0-cosor)^ 

 f x sin 2 ff sin 3 o- sin 2 a x 



tan^ , 2-tanJ? ^ ^ 0J0 _ tan gec 2 ffda) 

 r tan 2 o- r2 tan 3 <r 



2 sin 3 cos 07 „ A „ m 1 cos a , 

 = — (cos 0— cos 6)a<0= ^—dp,. 



cos 3 sin 2 a K ' cos 3 ri 



Now cos 0— cos o" or 2 sin i sin £' is positive ; and cos a is positive from 



t = Otoi = GT, and negative from i== to «=-. But (20) shows 



that ^ decreases as p increases. From z = to 2=sr, p, increases and 

 p 2 decreases, and therefore \p x decreases and ^increases, and therefore 



on both accounts x decreases. When 2= or, - 1 is still positive, and 



(.It 



therefore —p- negative, but \p 2 has its maximum value 1, so that on 



passing through the polarizing angle ^ still decreases, or the polari- 

 zation improves. When the plates are very numerous, \p 2 = 1 at the 

 polarizing angle, and on both sides of it decreases rapidly, whereas \p lt 

 which is always small, suffers no particular change about the polar- 

 izing angle. Hence in this case % must be a minimum a little beyond 

 the polarizing angle. Let us then seek the angle of incidence which 

 makes % a minimum in the case of an arbitrary number of plates. 

 We have from (20) and (2), 



_ sin 2 <t — sin 2 sin 2 a cos 2 + (2m — 1) sin 2 0cos 2 <r 



sin 2 a + ("2m — 1 ) sin 2 ' sin 2 cr cos 2 — sin 2 cos 2 <r 



_sin 2 <rcos 2 0+(2m — l)sin 2 0cos 2 o-_ , 2m 



sin 2 o- + (2m— l)sin 2 cosec 2 + (2m— 1) cosec 2 o-* ^ °' 



Hence % is a minimum along with cosec 2 + (2m — 1) cosec 2 <r. Differ- 

 entiating, and taking account of the formulae (22), we find, to deter- 



X 



