6 L. B. Tuckcnnan 



S, + , =J rQ k +cos2(£,£+i) sin2(£-4-i,£-f 2) sm2irN k+i 



+sin2(£,£+i) cos2(k—i,k-r2) cos27r7VJ fc+l sin2TrA^ +s 

 +sin2(£,£ + i) sin 2 ?riV r A . + x cos2 7ry\f +i! 



-X" A . +sin2(/^,^+i) sin2(^+ I ,^'-r 2 J sin2 7r.Y. .., 



— COS2(£,£+l) COS2(£+I,£-f 2) COS2 7rA^. + 1 sin2 7rX, _ .., 



— cos 2 (/£,£-[- 1 ) sin2 7rA^. +1 cos2 7rA^. +2 



+5J — cos 2(^+1,^— 2) sin27riV^ +1 sin27riV^ + , 

 1+ COS27riV s+1 COS27rA^ +! 



If in these equations the following substitutions are made : 



K k+i =K\ +1 S^,=S' k ^ JV k+2 =o 



they give the change produced by the (£-|-i)th plate when the 

 emergent light is referred to the same axes as the incident light, 

 instead of, as in the first form, to the axes of the (£-|-i)th plate. 

 This gives the theorem in its second form : 



P k + 1 =+P t 



& + i=+<&[i— "ran*?! k,k+i) shrViV, +1 ] 

 -\-K k sin 4(^,^+1) sin-'iri\^. +1 

 — S k sin2(£,£+i) sm2irJV k . rl 



K\ +1 =-)-Q % sm^k,k+ 1 ) sin'-VA^ 



-\-K k [i — 2 cos-2 (k, k-\- 1 ) sinV.Y + ,] 

 ~S k cos2(k,k- i r i) sin27riV i . +1 



'3) 



5' 1 ._ 1 =+Q A . sin 2(^,^+1) sin27rA r A . +1 



— K k COS2(k,fc+l) sin27rA'_ , 

 +S k COS2irN k+: 



This alternative form of the theorem, while not so simple 



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