Abstract

The Gaussian constants of a symmetrical optical instrument are built up from those of its parts by matrix multiplication. The matrix method is employed to find the constants for light which has been reflected any number of times between two given surfaces of the instrument. The matrix is expressible as the sum of two parts, each consisting of the product of a trigonometrical function depending on the number of reflections and a matrix which is independent of this number.

© 1929 Optical Society of America

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References

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  1. For a detailed discussion of this question with special reference to the signs of all the quantities involved see The treatment of reflection as a special case of refraction, Trans. Opt. Soc.,  27, p. 312; 1925–26.
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Trans. Opt. Soc. (1)

For a detailed discussion of this question with special reference to the signs of all the quantities involved see The treatment of reflection as a special case of refraction, Trans. Opt. Soc.,  27, p. 312; 1925–26.
[Crossref]

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Equations (27)

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( B D A C ) = ( b 1 d 1 a 1 c 1 ) + ( b 2 d 2 a 2 c 2 ) ,
A = a 1 + a 2 , B = b 1 + b 2 , C = c 1 + c 2 , D = d 1 + d 2 .
( W ) = ( w 1 ) + ( w 2 )
( W ) = ( w 1 ) ( w 2 ) ,
A = a 1 b 2 + c 1 a 2 , B = b 1 b 2 + d 1 a 2 , C = a 1 d 2 + c 1 c 2 , D = b 1 d 2 + d 1 c 2 ,
( b 2 d 2 a 2 c 2 ) ( b 1 d 1 a 1 c 1 ) = ( b 1 b 2 + a 1 d 2 d 1 b 2 + c 1 d 2 b 1 a 2 + a 1 c 2 d 1 a 2 + c 1 c 2 ) ,
a x x + b x - c x - d = 0
a 1 x x + b 1 x - c 1 x - d 1 = 0 a 2 x x + b 2 x - c 2 x - d 2 = 0
c 1 x + d 1 a 1 x + b 1 = x = - b 2 x - d 2 a 2 x - c 2
A x x + B x - C x - D = 0
( W ) = ( w 1 ) ( w 2 ) .
( 1 0 a 1 )
( 1 0 a 1 1 ) ( 1 0 a 2 1 ) ( 1 0 a 3 1 ) = ( 1 0 a 1 + a 2 + a 3 + 1 )
( 1 - X 0 1 ) ,
( 1 - X 0 1 ) .
( 1 0 a 1 1 ) ( 1 - t 0 1 ) ( 1 0 a 2 1 ) = ( 1 - a 2 t - t a 1 + a 2 - a 1 a 2 t 1 - a 1 t ) .
( 1 0 - μ i R i 1 ) .
( 1 0 μ i R i 1 ) ,
( w 1 ) ( w 2 ) ( w 2 ) ( w 2 ) ( w 2 ) ( w 2 ) ( w 3 )
( w 1 ) ( w 2 ) ( w 2 ) ( w 2 ) ( w 2 ) ( w 2 ) ( w 1 )
( p cos θ - q sin θ 1 q sin θ 1 p cos θ )
( p cos θ - q sin θ 1 q sin θ 1 p cos θ ) ( r p cos ϕ - q r sin ϕ r q sin ϕ p r cos ϕ ) = ( r cos ( θ + ϕ ) - p q r sin ( θ + ϕ ) r p q sin ( θ + ϕ ) 1 r cos ( θ + ϕ ) )
( r cos n θ - p q r sin n θ r p q sin n θ 1 r cos n θ )
( W ) = - sin ( n - 2 ) θ sin 2 θ ( W i ) + sin n θ sin 2 θ ( W j )
( W ) = - sin ( n - 3 ) θ sin 2 θ ( W 1 ) + sin ( n - 1 ) θ sin 2 θ ( W 3 )
cos ( n - 1 ) θ ( b 2 d 2 a 2 c 2 ) + 2 a 2 b 2 c 2 d 2 sin ( n - 1 ) θ sin 2 θ ( 1 c 2 1 a 2 1 d 2 1 b 2 )
sin ( n + 1 ) θ sin 2 θ ( b 2 d 2 a 2 c 2 ) - sin ( n - 1 ) θ sin 2 θ ( b 2 - d 2 - a 2 c 2 ) ,