## Abstract

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### Equations (37)

$F = ( n ′ − n ) R ,$
$α ′ − θ ′ = α − θ .$
$C · sin α n · sin θ = C ′ · sin α ′ n ′ · sin θ ′ = − R ;$
$n ′ · sin α ′ = n · sin α ,$
$n ′ 2 C ′ sin θ ′ = n 2 C sin θ .$
$E = ( sin θ ′ − sin α ′ ) − ( sin θ − sin α ) .$
$sin θ ′ sin θ = 1 + E sin θ + n n ′ · F C ,$
$n n ′ C ′ = n ′ n C ( 1 + E sin θ ) + F ;$
$C ′ n ′ 2 = C n 2 + F n n ′ − R n · sin α E .$
$α + α ′ + θ + θ ′ = 2 ( α ′ + θ ) , α − α ′ + θ − θ ′ = 2 ( α − α ′ ) ,$
$E = 4 sin α − θ 2 · sin α ′ + θ 2 · sin α ′ − α 2 .$
$n n ′ B ′ = n ′ n B + F .$
$p = sin α R = − n · sin θ C , p ′ = sin α ′ R = − n ′ · sin θ ′ C ′ , n ′ · p ′ = n · p .$
$C ′ n ′ 2 = C n 2 + F n · n ′ − E n · p .$
$C ′ k n k 2 + 1 = C k n k 2 + F k n k · n k + 1 − E k n k · p k , B ′ k n k 2 + 1 = B k n k 2 + F k n k · n k + 1 . }$
$C − B = δ C , C ′ − B ′ = δ C ′ ,$
$p k ′ 2 · δ C k ′ = p k 2 · δ C k − n k · p k · E k ;$
$δ C k + 1 = ( C k + 1 C k ′ ) 2 δ C k ′ , approximately ;$
$p k + 1 · C k + 1 = p k ′ · C k ′ ,$
$p k 2 + 1 · δ C k = p k ′ 2 · δ C k ′ , approximately .$
$p m ′ 2 · δ C m ′ = p 1 2 · δ C 1 − ∑ k = 1 k = m n k · p k · E k .$
$δ c ′ m = − n m + 1 δ C m ′ C m ′ 2 , approximately ,$
$δ c m ′ = n m + 1 ( p m ′ · C m ′ ) 2 ∑ k = 1 k = m n k · p k · E k , ( δ c 1 = 0 ) .$
$p k = − n k · sin θ k C k , p k ′ = n k n k + 1 p k , sin α k = p k · R k , sin α k ′ = p k ′ · R k , θ k + 1 = θ k + α k ′ − α k , C k ′ = − n k + 1 · sin θ k + 1 p k ′ , n k · p k · E k = ( n k · p k ) 2 ( F k n k · n k + 1 + C k n k 2 − C k ′ n k 2 + 1 ) . }$
$a k = d k + r k + 1 − r k ,$
$C k + 1 = n k + 1 · C k ′ n k + 1 − a k · C k ′ .$
$p k + 1 = p k ′ + a k · sin θ k + 1 ,$
$E k = sin θ k + 1 − sin α k ′ + sin α k − sin θ k ;$
$E k = 4 sin α k − θ k 2 sin α k ′ + θ k 2 sin α k ′ − α k 2 .$
$n 2 = n 4 = 1.61358 ( flint ) , n 3 = 1.51806 ( crown ) .$
$δ c 4 ′ = 0.0034 , approximately .$
$C 4 L 5 = 270.1185 ,$
$C 4 F ′ = 270.1167 ;$
$δ c 4 ′ = 0.0018 ,$
$1 n ′ ( B ′ n ′ + R ) = 1 n ( B n + R ) .$
$B n + R = − δ C n + C n + R = − δ C n + sin α − sin θ p ;$
$δ C ′ n ′ 2 − sin α ′ − sin θ ′ n ′ p ′ = δ C n 2 − sin α − sin θ n p .$