Doubly Refracting Plates and Elliptic Analysers 



'5 



origin along three right-handed rectangular axes, then the locus 

 P= constant, is a sphere. Poincare 1 has pointed out a beautiful 

 analogy between the action of a doubly refracting plate and the 

 rotation of a sphere. Examination shows that the sphere defined 

 above is the Poincare Sphere. 



The general linear transformation : 



Q, +1 =(^^. +1 )Q,-KA;,ft +1 )A:,-r(^^, +1 )5, 



K k+ ^(Q v K k ^)Q,A-{K v K k+l )K k +{S v K k+l )S k (4) 



S k+l =( O k ,S k+l )Q k +(K k , S k+l )K k +(S v S k+1 )S k 



represents the rotation of a solid body, provided : 



(Q k ,Q k+1 y+(K k ,Q k+1 y+(s k ,o k+1 y=i 

 (o v K k+l y+{K k ,K k+l y+{s k ,K k+1 y=i (26) 



(Q k , s k+1 y+(xr k , s k ^y+(s k , s k+l y=i 



{Q k ,A' k + l ){O k , S k ^ + {K v K k + l )(K k , S k ^) + {S k ,K k+l ){S v S k+l )=o 

 (^^.JC^^.J + C^^.OC/^^.J + C^^./jC^^^Ho 



(27) 



and 



(Q k ,Q k+l ), (a;^. +1 ), (s k ,o k+1 ) 

 (Q k) K k+1 ), (KK +l ), (<s»K +l ) 



H-i 



(28) 



These equations are satisfied by both forms of the inductive 

 and general theorems given above. The coefficients of a linear 

 transformation satisfying the above conditions may be written : 



( Qv Qk+\^ = l — 2sin 2 rt sin 2 j4 r 

 (A" 4 ,A" fc+1 ) = i — 2sin 2 £sin 2 ^r 

 (S k , S k+1 )=l — 2sin 2 ^sin 2 ^r 



(29) 



Poincare H. Theorie Mathematique de la Lumierc, Paris, 1892, T. 2, 

 pp. 275-85. 



171 



