Doubly Refracting Plates and Elliptic Analyzers 7 



and therefore not usually so convenient as the first, is useful in 

 discussing the effect of the rotation of a single plate, as in the 

 discussion of errors and in the geometrical analogy of Poincare's 

 sphere. 



2. General Theorem 



To obtain the effect of 11 successive plates it is necessary to ap- 

 ply the inductive theorem u times in succession. The transfor- 

 mations of Q, K and Sin equations (2) and (3) are linear trans- 

 formations. A series of linear transformations give a linear 

 transformation as a resultant, the coefficients of the resultant 

 transformation being obtained from those of the series by the 

 theorem for the multiplication of determinants. The following 

 notation is convenient : 



Q k+ i=(Q„Q* +1 )Q k +(KQ> +1 )K«+(s*>Q*+i)S k 



K k ^=iQ v K k+ AQ k ^{K v K k+l )K k +(S v K k+l )S k (4) 

 S k ^={Q k ,S k+l )Q k Ar{K< S k+1 )K k +(S v S k+1 )S k 

 fc=o, 1,2,3, • • • -i n — O- 

 and 



Q= Q , Qj Q +(K> QJ K ^-(S , Q n )S 



K=(Q ,K) Q.+{K#JK.+(S*iOS. (5) 



SMQ* SJQo+tK Sn)K +(S , S n )S 



where ( O v A^ +1 ) is a symbol representing the coefficient of Q k in 



the expression for K k+V (^Q,,) representing the coefficient of S in 



Q n , etc. 



The effect of ?i successive plates is then known when the co- 

 efficients (Q ,OJ, (Q n ,A'J, etc., are expanded in terms of 

 (Q k ,O k _J, (Q k ,J? k+1 ), etc., for £=0,1,2,3, • • • • ( n ~ *)■ The 

 theorem for the multiplication of determinants gives: 



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