8 L. B. Tuckcrman 



where a k =Q k ,K k , or S k , and jL^ indicates the summation of the 



results obtained by substituting these values in all possible combi- 

 nations. 



This is the required relation between (Q^OJ, (Q 0% Jf n ), etc., and 



(Q k ,g k+1 ), (&*;„), etc. 



The coefficients (a A .,a ;+1 ) lor the two forms of the inductive 

 theorem are: 



First Form 



(£*,£* +1 )=+cos2C£,/H-i) 



( A' A , A' i _ ( . 1 )= + COS2(£,X* + I ) COS2 7T A, + , 



( A", 6" A + 1 )= — cos2(X\X'— i ) sin 2 7riY ; i+] 



(6;, A; +1 )=+sin27r^ + 1 (7) 



(6-,, £ A+1 )=o 



( £> A ., 6" i + j ) =+sin 2 ( k, /fc-f- 1 ) sin 2 ttA^ _ , 



(Q k , A^ +1 )= — sin2(X',X- L i ) eos277,Y ; _ ] 



(A' A , <9, +1 )=+sin 2 (X',X+i) 



Second Form 



(£,, £, +1 )=i-2sin 2 2( X,X-i) sinVA^, 



(A;, A; + 1 ) = i— 2cos" 2 2(X-,XM-i) sin^yV^j 



(•Si*. ^ +1 )=+ cos ? 7r ^+i 



( X\" , S k + l ) = — cos 2 ( k, k-\- 1 ) sin 2 tt A 7 ';. + , 



(S v A; + 1 )=+cos2(X',X+i)sin 2 7rA; +1 (8) 



( -5"*. £?* + 1 ) =— sin 2 ( X, X-+ 1 ) sin 2 tt A] + , 

 (£*, 5, +1 )=+sin2(X,X'+i) sin27rA^, +1 

 (P,.A; +1 )=+sin 4 (^X+i) sin 2 irAf +1 

 (A' A , 0^ 1 )=+sin4(X,X+i) shrVA,', 



The- equations for the effect of any number of plates can then be 

 written directly in either of the two forms by the use of equation 



164 



