Abstract

An image error theory is developed for finite aperture and field. In the limit the formulas reduce to those of the well-known Seidel theory. The constants of the Seidel theory are replaced in the general theory by functions. It is shown that for a given object plane these functions can be reduced to two independent functions, the vanishing of one leading to a symmetrical image, the vanishing of both to a sharp image, and it also is proved that these two functions permit one to compute the caustic surfaces. The image error functions for a given plane can be identified as the second-order derivatives of a characteristic function. To ascertain the image quality for a different plane, transformation formulas are developed which permit one to compute the first- and second-order derivatives of this characteristic function for any shift of the object plane. The authors have found that the second-order derivatives of another characteristic function are invariant with respect to a shift of the object and that a complete system of invariants of the image errors can be obtained. The basic formulas demonstrate that the shifted image-errors are linear functions of the image errors of the original plane and their second- and third-order determinants. Applications to classical problems are given.

© 1952 Optical Society of America

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  1. T. Smith, Trans. Opt. Soc. (London) 23, 311 (1921–1922).
    [Crossref]
  2. M. Herzberger, J. Opt. Soc. Am. 38, 736 (1948).
    [Crossref] [PubMed]

1948 (1)

Herzberger, M.

Smith, T.

T. Smith, Trans. Opt. Soc. (London) 23, 311 (1921–1922).
[Crossref]

J. Opt. Soc. Am. (1)

Trans. Opt. Soc. (London) (1)

T. Smith, Trans. Opt. Soc. (London) 23, 311 (1921–1922).
[Crossref]

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Equations (114)

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ξ 2 + η 2 + ζ 2 = μ 2 .
ξ 2 + η 2 + ζ 2 = μ 2 ,
u = 1 2 ( x 2 + y 2 ) v = x ξ + y η w = ζ .
a = ζ ,             b = ξ ξ + η η ,             c = ζ .
ξ = - V 1 x - V 2 ξ x = - V 2 x + ( V 3 ξ / c ) η = - V 1 y - V 2 η y = - V 2 y + ( V 3 η / c ) ,
μ 2 - a 2 = 2 u V 1 2 + 2 v V 1 V 2 + ( μ 2 - w 2 ) V 2 2 b = - V 1 v - V 2 ( μ 2 - w 2 ) c = w ,
x = - ( W 1 ξ / a ) + W 2 ξ x = - W 2 ξ + ( W 3 ξ / c ) y = - ( W 1 η / a ) + W 2 η y = - W 2 η + ( W 3 η / c )
2 u = W 1 2 / a 2 ( μ 2 - a 2 ) - ( 2 W 1 W 2 b / a ) + W 2 2 ( μ 2 - c 2 ) v = - ( W 1 b / a ) + W 2 ( μ 2 - c 2 ) w = c .
x D = - V 2 x y D = - V 2 y z D = - V 3 .
V 2 = f ( V 3 )
V 22 V 33 - V 23 2 0 ,
x D x = f ( z ) = y D y .
k 2 μ 2 - k B + C = 0 ,
B = V 22 ( 2 u μ 2 - V 2 ) - 2 v w V 23 + V 33 ( μ 2 - w 2 ) C = D ( V 22 V 33 - V 23 2 )
D = 2 u ( μ 2 - w 2 ) - v 2 .
V 1 = a / W 1 W 1 = a / V 1 V 2 = - a W 2 / W 1 W 2 = - V 2 / V 1 V 3 = W 3 - ( a c W 2 2 / W 1 ) W 3 = V 3 + ( V 2 2 w / V 1 ) .
W ¯ ( z ) = W ( 0 ) - a z ,
V ¯ 1 = a W ¯ 1 = a W 1 - z = a ( a / V 1 ) - z = V 1 1 - ( V 1 z / a ) V ¯ 2 = - a W ¯ 2 W ¯ 1 = - a W 2 W 1 - z = V 2 1 - ( V 1 z / a ) V ¯ 3 = V 3 - w V 2 2 ( z / a ) 1 - V 1 ( z / a ) .
2 ū = 2 u - 2 z a ( 2 u V 1 + v V 2 ) + z 2 a 2 ( 2 u V 1 2 + 2 v V 1 V 2 + V 2 2 / ( μ 2 - w 2 ) ) v ¯ = v - z a ( V 1 v + V 2 ( μ 2 - w 2 ) ) w ¯ = w
μ 2 - a 2 = 2 u V 1 2 + 2 v V 1 V 2 + V 2 2 / ( μ 2 - w 2 ) .
u = v = w 2 - μ 2 = a 2 - μ 2 = b = c 2 - μ 2 = 0.
x D = - V 2 x y D = - V 2 y z D = - V 3
k 2 μ 2 - k [ V 22 ( 2 u μ 2 - v 2 ) - 2 v w V 23 + V 33 ( μ 2 - w 2 ) ] + D ( V 22 V 33 - V 2 23 ) = 0
x D = y D = 0 z D = - V 3 ,
ξ = - V 2 ξ η = - V 2 η .
x D = - V 2 x y D = - V 2 y z D = - V 3 ,
k 2 μ 2 - k V 22 2 u μ 2 = 0 ,
( V 11 V 12 - 1 c V 13 V 12 V 22 - 1 c V 23 - 1 c V 13 - 1 c V 23 1 c 2 V 33 ) , ( W 11 W 12 W 13 W 12 W 22 W 23 W 13 W 23 W 33 ) ,
V 1 = a / W 1 W 1 = a / V 1 V 2 = - a W 2 / W 1 W 2 = - V 2 / V 1 V 3 = W 3 - ( a c W 2 2 / W 1 ) W 3 = V 3 + ( V 2 2 c / V 1 ) .
μ 2 - a 2 = 2 u V 1 2 + 2 v V 1 V 2 + V 2 2 ( μ 2 - w 2 ) b = - V 1 v - V 2 ( μ 2 - w 2 ) c = w ,
2 u = μ 2 - a 2 a 2 W 1 2 - 2 b a W 1 W 2 + ( μ 2 - c 2 ) W 2 2 v = - b a W 1 + ( μ 2 - c 2 ) W 2 w = c .
A = ( u a u b u c v a v b v c w a w b w c )             and             ( a u a v a w b u b v b w c u c v c w ) = B = A - 1 ,
d u = ( W 1 α 11 + W 2 α 21 ) d a + ( W 1 α 12 + W 2 α 22 ) d b + ( W 1 α 13 + W 2 α 23 ) d c d v = α 21 d a + α 22 d b + α 23 d c d w = d c ,
α 11 = - μ 2 a 3 W 1 + μ 2 - a 2 a 2 W 11 - b a W 12 α 12 = μ 2 - a 2 a 2 W 12 - b a W 22 α 13 = c W 2 2 W 1 + μ 2 - a 2 a 2 W 13 - b a W 23 α 21 = b a 2 W 1 - b a W 11 + ( μ 2 - c 2 ) W 12 α 22 = - 1 a W 1 - b a W 12 + ( μ 2 - c 2 ) W 22 α 23 = - 2 c W 2 - b a W 13 + ( μ 2 - c 2 ) W 23 .
Δ = W 1 | α 11 α 12 α 21 α 22 | = W 1 [ ( W 1 a ) 2 μ 2 a 2 - W 1 a ( μ 2 - a 2 a 2 W 11 - 2 b a W 12 + ( μ 2 - c 2 + δ ) W 22 ) + δ q 33 ] ,
δ = ( ( μ 2 - a 2 ) ( μ 2 - c 2 ) - b 2 ) / a 2 .
A = 1 Δ ( α 22 , - ( W 1 α 12 + W 2 α 22 ) , W 1 ( α 12 α 23 - α 22 α 13 ) - α 21 , W 1 α 11 + W 2 α 21 , - W 1 ( α 11 α 23 - α 21 α 13 ) 0 , 0 , W 1 ( α 11 α 22 - α 21 α 12 ) ) .
ω 11 = - 1 a + 1 W 1 ( W 11 + W 22 ( μ 2 - c 2 ) ) - a W 1 2 ( μ 2 - c 2 ) q 33 ω 12 = W 12 + W 22 b a - b a W 1 a q 33 ω 22 = W 1 W 22 μ 2 a 2 - μ 2 - a 2 a 2 a q 33 ω 13 = 1 c [ W 1 a ( W 13 + b a W 23 ) + b a q 23 + ( μ 2 - c 2 ) q 13 ] ω 23 = 1 c W 1 [ W 1 a W 23 μ 2 a 2 + μ 2 - a 2 a 2 q 23 + b a q 13 ] ω 33 = 1 c 2 { W 1 3 μ 2 a 4 W 33 - W 1 2 a [ W 2 2 μ 2 a 2 + μ 2 - a 2 a 2 q 22 + 2 b a q 12 + ( μ 2 - c 2 + δ ) q 11 ] + W 1 [ W 2 2 ( μ 2 - a 2 a 2 W 11 - 2 b a W 12 + ( μ 2 - c 2 + δ ) W 22 ) + δ q 123 ] - a δ W 2 2 q 33 } .
V 11 Δ = ω 11 V 12 Δ = - W 2 ω 11 + ω 12 - 1 c V 13 Δ = W 2 2 ω 11 - 2 W 2 ω 12 + ω 13 V 22 Δ = W 2 2 ω 11 - 2 W 2 ω 12 + ω 22 - 1 c V 23 Δ = - W 2 3 ω 11 + 3 W 2 2 ω 12 - W 2 ( ω 13 + 2 ω 22 ) + ω 23 1 c 2 V 33 Δ = W 2 4 ω 11 - 4 W 2 3 ω 12 + 2 W 2 2 ( ω 13 + 2 ω 22 ) - 4 W 2 ω 33 + ω 33 .
p 11 Δ = W 2 4 π 33 + 2 W 2 3 π 23 + W 2 2 ( π 22 + 2 π 13 ) + 2 W 2 π 12 + π 11 p 22 Δ = 4 W 2 2 π 33 + 4 W 2 π 23 + π 22 p 12 Δ = 2 W 2 3 π 33 + 3 W 2 2 π 23 + W 2 ( π 22 + 2 π 13 ) + π 12 p 23 Δ = 2 W 2 π 33 + π 23 p 13 Δ = W 2 2 π 33 + W 2 π 23 + π 13 p 33 Δ = π 33 .
p 123 Δ = π 123 .
γ 11 = W 11 - ( W 1 / a ) γ 22 = W 22 γ 12 = W 12 γ 23 = W 23 γ 13 = W 13 γ 33 = W 33 - ( W 2 2 a / W 1 )
Γ 11 = q 11 - a W 2 2 W 1 W 22 Γ 22 = q 22 - W 1 a W 33 - a W 2 2 W 1 W 11 + W 2 2 Γ 12 = q 12 + a W 2 2 W 1 W 12 Γ 23 = q 23 + W 1 a W 23 Γ 13 = q 13 Γ 33 = q 33 - W 1 a W 22 Γ 123 = q 123 - a W 2 2 W 1 q 33 - W 1 a q 11 + W 2 2 W 22 .
ω 11 = 1 W 1 γ 11 - a W 1 2 ( μ 2 - c 2 ) Γ 33 ω 12 = γ 12 - b W 1 Γ 33 ω 22 = W 1 γ 22 - μ 2 - a 2 a Γ 33 ω 13 = 1 c [ W 1 a γ 13 + b a Γ 23 + ( μ 2 - c 2 ) Γ 13 ] ω 23 = W 1 c [ W 1 a γ 23 + μ 2 - a 2 a 2 Γ 23 + b a Γ 13 ] ω 33 = a c 2 { ( W 1 a ) 3 γ 33 - ( W 1 a ) 2 [ μ 2 - a 2 a 2 Γ 22 + 2 b a Γ 12 + ( μ 2 - c 2 ) Γ 11 ] + δ W 1 a Γ 123 } .
π 11 = W 1 c 2 ( Γ 11 - μ 2 - a 2 a W 1 Γ 123 ) π 13 = a c W 1 Γ 13 π 12 = 1 c 2 ( Γ 12 + b W 1 Γ 123 ) π 23 = a c W 1 2 Γ 23 π 22 = 1 c 2 W 1 ( Γ 22 - a ( μ 2 - c 2 ) W 1 Γ 123 ) π 33 = a 2 W 1 3 Γ 33 π 123 = a 2 c 2 W 1 3 Γ 123 .
= V 1 a [ V 1 2 + V 1 ( 2 V 11 u + 2 V 12 v + V 22 ( μ 2 - w 2 ) ) + D p 33 ] .
ω 11 = V 11 ω 12 = 1 V 1 ( V 1 V 12 - V 2 V 11 ) ω 22 = 1 V 1 2 ( V 1 2 V 22 - 2 V 1 V 2 V 12 + V 2 2 V 11 ) ω 13 = 1 V 1 2 ( V 1 2 ( - V 13 w ) - 2 V 1 V 2 V 12 + V 2 2 V 11 ) ω 23 = 1 V 1 3 [ V 1 3 ( - V 23 w ) - V 1 2 V 2 ( 2 V 22 - V 13 w ) + 3 V 1 V 2 2 V 12 - V 2 3 V 11 ] ω 33 = 1 V 1 4 [ V 1 4 ( V 33 w 2 ) - 4 V 1 3 V 2 ( - V 23 w ) + 2 V 1 2 V 2 2 ( 2 V 22 - V 13 w ) - 4 V 1 V 2 3 V 12 + V 2 4 V 11 .
π 11 = p 11 + 2 V 2 V 1 p 12 + ( V 2 V 1 ) 2 ( 2 p 13 + p 22 ) + 2 ( V 2 V 1 ) 3 p 23 + ( V 2 V 1 ) 4 p 33 π 12 = p 12 + V 2 V 1 ( 2 p 13 + p 22 ) + 3 ( V 2 V 1 ) 2 p 23 + 2 ( V 2 V 1 ) 3 p 33 π 13 = p 13 + V 2 V 1 p 23 + ( V 2 V 1 ) 2 p 33 π 22 = p 22 + 4 V 2 V 1 p 23 + 4 ( V 2 V 1 ) 2 p 33 π 23 = p 23 + V 2 V 1 p 33 , π 33 = p 33 , π 123 = p 123 .
γ 11 = a V 1 ( ω 11 + μ 2 - c 2 V 1 π 33 ) , γ 13 = c V 1 ( ω 13 - b V 1 2 π 23 - μ 2 - c 2 V 1 π 13 ) γ 12 = ( ω 12 + b V 1 2 π 33 ) , γ 23 = c V 1 2 a ( ω 33 - μ 2 - a 2 V 1 3 - b V 1 2 π 13 ) γ 22 = V 1 a ( ω 22 + μ 2 - a 2 V 1 3 π 33 ) , γ 33 = c 2 V 1 3 a ( ω 33 + 1 V 1 ( μ 2 - c 2 ) π 11 + 2 b V 1 2 π 12 + μ 2 - a 2 V 1 3 π 22 + δ a 2 V 1 4 π 123 )
Γ 11 = c 2 V 1 a ( π 11 + μ 2 - a 2 V 1 3 π 123 ) Γ 13 = c V 1 π 13 π 12 = c 2 ( π 12 - b V 1 2 π 123 ) Γ 23 = a c V 1 2 π 23 π 22 = a c 2 V 1 ( π 22 + μ 2 - c 2 V 1 π 123 ) Γ 33 = a V 1 3 π 33 , Γ 123 = a c 2 V 1 3 π 123 .
W 11 = γ 11 + 1 V 1 W 22 = γ 22 W 23 = γ 23 W 12 = γ 12 W 13 = γ 13 W 33 = γ 33 + V 2 2 V 1
q 11 = Γ 11 + V 2 2 V 1 γ 22 q 22 = Γ 22 + 1 V 1 γ 33 + V 2 2 V 1 γ 11 + V 2 2 V 1 2 q 12 = Γ 12 - V 2 2 V 1 γ 12 q 23 = Γ 23 - 1 V 1 γ 23 q 13 = Γ 13 q 33 = Γ 33 + 1 V 1 γ 22 q 123 = Γ 123 + 1 V 1 Γ 11 + V 2 2 V 1 Γ 33 + V 2 2 V 1 2 γ 22 .
μ 2 k 2 - k B + C = 0 ,
B = V 22 ( 2 u μ 2 - v 2 ) - 2 v w V 23 + V 33 ( μ 2 - w 2 ) = V 22 D + w 2 ( 2 u V 22 + 2 v ( - V 23 w ) + ( μ 2 - w 2 ) V 33 w 2 ) C = w 2 D p 11 .
B Δ = D K 1 + w 2 V 1 2 ( ( μ 2 - a 2 ) K 1 - 2 b K 2 V 1 + ( μ 2 - c 2 ) K 3 V 1 2 ) C Δ = ω 2 D ( K 1 K 3 - K 2 2 ) ,
K 1 = ω 22 + 2 V 2 V 1 ω 12 + ( V 2 V 1 ) 2 ω 11 K 2 = ω 23 + V 2 V 1 ( ω 22 + ω 13 ) + ( V 2 V 1 ) 2 ω 12 K 3 = ω 33 + 2 V 2 V 1 ω 23 + ( V 2 V 1 ) 2 ω 22 .
K 1 K 3 - K 2 2 = 0
K 1 K 3 - K 2 2 = p 11 Δ ,
K 1 = K 2 = K 3 = 0.
K 1 = K 3 = p 11 = 0 ,
W ¯ = W - a z ,
W ¯ 1 = W 1 - z W ¯ i k = W i k W ¯ 2 = W 2 q ¯ i k = q i k . W ¯ 3 = W 3
V ¯ 1 = a W 1 - z = V 1 1 - ( V 1 z / a ) V ¯ 3 = V 3 - w V 2 2 ( z / a ) 1 - ( V 1 z / a ) . V ¯ 2 = - a W 2 W 1 - z = V 2 1 - ( V 1 z / a )
τ = 1 1 - ( V 1 z / a ) ,
V ¯ 1 = τ V 1 ,             V ¯ 2 = τ V 2 ,             τ - 1 = τ V 1 z / a .
γ ¯ 11 = - 1 V 1 τ + ( γ 11 + 1 V 1 ) γ ¯ 22 = γ 22 γ ¯ 23 = γ 23 γ ¯ 12 = γ 12 γ ¯ 13 = γ 13 γ ¯ 33 = ( γ 33 + V 2 2 V 1 ) - V 2 2 τ V 1 ,
Γ ¯ 11 = ( Γ 11 + V 2 2 V 1 γ 22 ) - ( V 1 τ ) V 2 2 V 1 2 γ 22 , Γ ¯ 22 = - 1 V 1 τ ( γ 33 + V 2 2 V 1 ) + ( Γ 22 + 1 V 1 γ 33 + V 2 2 V 1 γ 11 + 2 V 2 2 V 1 2 ) - ( V 1 τ ) V 2 2 V 1 2 ( γ 11 + 1 V 1 ) Γ ¯ 12 = ( Γ 12 - V 2 2 V 1 γ 12 ) + ( V 1 τ ) V 2 2 V 1 2 γ 12 , Γ ¯ 23 = 1 V 1 τ γ 23 + ( Γ 23 - 1 V 1 γ 23 ) Γ ¯ 13 = Γ 13 , Γ ¯ 33 = - 1 V 1 τ γ 22 + ( Γ 33 + 1 V 1 γ 22 ) Γ ¯ 123 = - 1 V 1 τ ( Γ 11 + V 2 2 V 1 γ 22 ) + ( Γ 123 + 1 V 1 Γ 11 + V 2 2 V 1 Γ 33 + 2 V 2 2 V 1 2 γ 22 ) - ( V 1 τ ) V 2 2 V 1 2 ( Γ 33 + 1 V 1 γ 22 ) .
ω 11 = 1 a [ - 1 + V 1 τ ( γ 11 + 1 V 1 + γ 22 ( μ 2 - c 2 ) ) - ( V 1 τ ) 2 ( μ 2 - c 2 ) ( Γ 33 + 1 V 1 γ 22 ) ] ω 12 = γ 12 + b a γ 22 - V 1 τ b a ( Γ 33 + 1 V 1 γ 22 ) ω 22 = a [ 1 V 1 τ μ 2 a 2 γ 22 - μ 2 - a 2 a 2 ( Γ 33 + 1 V 1 γ 22 ) ] ω 13 = 1 c [ 1 V 1 τ ( γ 13 + b a γ 23 ) + b a ( Γ 23 + 1 V 1 γ 23 ) + ( μ 2 - c 2 ) Γ 13 ] ω 23 = a c { ( 1 V 1 τ ) 2 μ 2 a 2 γ 23 + 1 V 1 τ [ μ 2 - a 2 a 2 ( Γ 23 - 1 V 1 γ 23 ) + b a Γ 13 ] } ω 33 = a c 2 { ( 1 V 1 τ ) 3 μ 2 a 2 ( γ 33 + V 2 2 V 1 ) - ( 1 V 1 τ ) 2 [ V 2 2 V 1 2 + μ 2 - a 2 a 2 ( Γ 22 + 1 V 1 γ 33 + V 2 2 V 1 γ 11 + 2 V 2 2 V 1 2 ) + 2 b a ( Γ 12 - V 2 2 V 1 γ 12 ) + ( μ 2 - c 2 + δ ) ( Γ 11 + V 2 2 V 1 γ 22 ) ] + 1 V 1 τ [ V 2 2 V 1 2 ( μ 2 - a 2 a 2 ( γ 11 + 1 V 1 ) - 2 b a γ 12 + ( μ 2 - c 2 ) γ 22 ) + δ ( Γ 123 + 1 V 1 Γ 11 + V 2 2 V 1 Γ 33 + 2 V 2 2 V 1 2 γ 22 ) ] - δ V 2 2 V 1 2 ( Γ 33 + 1 V 1 γ 22 ) } ,
π 11 = a c 2 { 1 V 1 τ μ 2 a 2 ( Γ 11 + V 2 2 V 1 γ 22 ) - [ V 2 2 V 1 2 γ 22 + μ 2 - a 2 a 2 ( Γ 123 + 1 V 1 Γ 11 + V 2 2 V 1 Γ 33 + 2 V 2 2 V 1 2 γ 22 ) ] + V 1 τ μ 2 - a 2 a 2 V 2 2 V 1 2 ( Γ 33 + 1 V 1 γ 22 ) } π 12 = 1 c 2 { [ Γ 12 - V 2 2 V 1 γ 12 - b a ( Γ 11 + V 2 2 V 1 γ 22 ) ] + V 1 τ [ V 2 2 V 1 2 γ 12 + b a ( Γ 123 + 1 V 1 Γ 11 + V 2 2 V 1 Γ 33 + 2 V 2 2 V 1 2 γ 22 ) ] - ( V 1 τ ) 2 b a V 2 2 V 1 2 ( Γ 33 + 1 V 1 γ 22 ) } π 22 = 1 a c 2 { - ( γ 33 + V 2 2 V 1 ) + V 1 τ [ Γ 22 + 1 V 1 γ 33 + V 2 2 V 1 γ 11 + 2 V 2 2 V 1 2 + ( μ 2 - c 2 ) ( Γ 11 + V 2 2 V 1 γ 22 ) ] - ( V 1 τ ) 2 [ ( μ 2 - c 2 ) ( Γ 123 + 1 V 1 Γ 11 + V 2 2 V 1 Γ 33 + 2 V 2 2 V 1 2 γ 22 ) + V 2 2 V 1 2 ( γ 11 + 1 V 1 ) ] + ( V 1 τ ) 3 ( μ 2 - c 2 ) V 2 2 V 1 2 ( Γ 33 + 1 V 1 γ 22 ) } π 13 = 1 c V 1 τ Γ 13 π 23 = 1 a c { V 1 τ γ 23 + ( v 1 τ ) 2 ( Γ 23 - 1 V 1 γ 23 ) } π 33 = 1 a { - ( V 1 τ ) 2 γ 22 + ( V 1 τ ) 3 ( Γ 33 + 1 V 1 γ 22 ) } π 123 = 1 a c 2 { - ( V 1 τ ) 2 ( Γ 11 + V 2 2 V 1 γ 22 ) + ( V 1 τ ) 3 ( Γ 123 + 1 V 1 Γ 11 + V 2 2 V 1 Γ 33 + 2 V 2 2 V 1 2 γ 22 ) - ( V 1 τ ) 4 V 2 2 V 1 2 ( Γ 33 + 1 V 1 γ 22 ) } .
p 11 Δ ¯ = 1 a c 2 1 V 2 τ { ( V 2 τ ) 4 μ 2 V 2 V 1 q 33 - ( V 2 τ ) 3 [ ( μ 2 - c 2 ) q 123 - 2 V 2 V 1 ( b q 33 - c q 23 ) + V 2 2 V 1 2 ( W 11 + μ 2 W 22 ) ] - ( V 2 τ ) 2 [ 2 b q 123 - V 2 V 1 [ ( μ 2 - a 2 ) q 33 + 2 a c q 13 + q 22 + ( μ 2 - c 2 ) q 11 ] + 2 V 2 2 V 1 2 ( a W 12 + b W 22 + c W 23 ) - V 2 3 V 1 3 ] - V 2 τ [ ( μ 2 - a 2 ) q 123 - 2 V 2 V 1 ( b q 11 - a q 12 ) + V 2 2 V 1 2 ( W 33 + μ 2 W 22 ) ] + μ 2 V 2 V 1 q 11 } .
K 1 = 1 a V 2 τ { - ( V 2 τ ) 3 ( μ 2 - c 2 ) q 33 - ( V 2 τ ) 2 [ 2 b q 33 - V 2 V 1 ( W 11 + ( μ 2 - c 2 ) W 22 ] - V 2 τ [ ( μ 2 - a 2 ) q 33 - 2 V 2 V 1 ( a W 12 + b W 22 ) + V 2 2 V 1 2 ] + V 2 V 1 μ 2 W 22 } K 2 = 1 a c ( V 2 τ ) 2 V 2 V 1 { - ( V 2 τ ) 3 b c q 33 + ( V 2 τ ) 2 [ a ( μ 2 - c 2 ) q 13 + b q 23 - c ( μ 2 - a 2 ) q 33 + V 2 c V 1 ( a W 12 + b W 22 ) ] + V 2 τ [ ( μ 2 - a 2 ) q 23 + a b q 13 + V 2 V 1 ( μ 2 c W 22 + a W 13 + b W 23 ) ] + V 2 V 1 μ 2 W 23 } K 3 = a c 2 ( V 2 τ ) 3 { V 2 3 V 1 3 μ 2 a 2 W 33 - ( V 2 τ ) V 2 2 V 1 2 [ μ 2 a 2 V 2 2 V 1 2 - 2 c V 2 V 1 μ 2 a 2 W 23 + μ 2 - a 2 a 2 q 22 + 2 b a q 12 + ( μ 2 - c 2 + δ ) q 11 ] + ( V 2 τ ) 2 V 2 V 1 [ V 2 2 V 1 2 ( μ 2 - a 2 a 2 W 11 - 2 b a W 12 + ( μ 2 - c 2 + δ + μ 2 c 2 a 2 ) W 22 ) + 2 c V 2 V 1 ( μ 2 - a 2 a 2 q 23 + b a q 13 ) + δ q 123 ] + ( V 2 τ ) 3 [ - V 2 2 V 1 2 q 33 ( c 2 μ 2 - a 2 a 2 + δ ) ] } .
V 22 = V 33 = p 11 = 0.
V 22 τ Δ ¯ Δ a 2 = V 22 a 2 + 1 - τ V 1 [ - V 1 2 V 2 2 - 2 V 2 V 12 ( μ 2 - v V 1 V 2 - ( μ 2 - w 2 ) V 2 2 ) + V 1 V 22 ( 2 u V 1 2 - ( μ 2 - w 2 ) V 2 2 ) + p 33 ( 2 u μ 2 - D V 2 2 ) ] + ( 1 - τ V 1 ) 2 V 2 [ V 1 3 V 2 + V 2 V 11 ( μ 2 - ( μ 2 - w 2 ) V 2 2 ) - 2 v V 1 2 V 2 V 12 + V 1 2 V 22 ( 2 v V 1 + ( μ 2 - w 2 ) V 2 ) + p 33 ( 2 v μ 2 + D V 1 V 2 ) ] + ( 1 - τ V 1 ) 3 V 2 2 ( μ 2 - w 2 ) [ V 1 ( V 2 2 V 11 - 2 V 1 V 2 V 12 + V 1 2 V 22 ) + μ 2 p 33 ]
a 2 = μ 2 - 2 u V 1 2 - 2 v V 1 V 2 - ( μ 2 - w 2 ) V 2 2 - V 23 w τ 2 Δ ¯ Δ a 2 = - V 23 w a 2 + 1 - τ V 1 [ - V 1 V 2 3 - V 2 2 V 12 ( 2 u V 1 + v V 2 ) + V 2 V 22 ( - 2 μ 2 + 4 u V 1 2 V 2 + 3 v V 1 V 2 2 + ( μ 2 - w 2 ) V 2 3 ) + V 2 ( - V 13 w ) ( - μ 2 + v V 1 V 2 + ( μ 2 - w 2 ) V 2 2 ) + V 1 2 ( - V 23 w ) ( 2 u V 1 + v V 2 ) + p 13 ( v μ 2 + D V 1 V 2 ) + p 23 ( - 2 u μ 2 + D V 2 2 ) ] + ( 1 - τ V 1 ) 2 [ 2 V 1 2 V 2 3 + v V 2 4 V 11 + V 2 2 V 12 ( 3 μ 2 + 2 u V 1 2 - 3 v V 1 V 2 - 2 ( μ 2 - w 2 ) V 2 2 ) + V 1 V 2 V 22 ( - 4 u V 1 2 + ( μ 2 - w 2 ) V 2 2 ) - V 1 V 2 2 ( - V 13 w ) ( V 1 v + V 2 ( μ 2 - w 2 ) ) + V 1 2 V 2 ( - V 23 w ) ( V 1 v + V 2 ( μ 2 - w 2 ) ) + V 2 p 13 ( μ 2 ( μ 2 - w 2 ) - D V 1 2 ) - V 2 p 23 ( v μ 2 + D V 1 V 2 ) + V 2 p 33 ( - 4 u μ 2 + D V 2 2 ) ] + ( 1 - τ V 1 ) 3 [ - V 1 3 V 2 3 + V 2 3 V 11 ( - μ 2 - v V 1 V 2 + ( μ 2 - w 2 ) V 2 2 ) + 4 v V 1 2 V 2 3 V 12 - 3 V 1 2 V 2 2 V 22 ( V 1 v + V 2 ( μ 2 - w 2 ) ) - V 2 2 p 33 ( 3 v μ 2 + D V 1 V 2 ) ] + ( 1 - τ V 1 ) 4 ( - V 2 3 ) ( μ 2 - w 2 ) [ V 1 ( V 2 2 V 11 - 2 V 1 V 2 V 12 + V 1 2 V 22 ) + μ 2 p 33 ]
V 33 w 2 τ 3 Δ ¯ Δ a 2 = V 33 a 2 w 2 + 1 - τ V 1 [ V 2 2 w 2 ( μ 2 - 2 u V 1 2 - 2 v V 1 V 2 - μ 2 V 2 2 ) - 2 V 2 2 ( - V 13 w ) ( 2 u V 1 + v V 2 ) + 2 V 2 ( - V 23 w ) ( - 2 μ 2 + 4 u V 1 2 + 3 v V 1 V 2 + ( μ 2 - w 2 ) V 2 2 ) + V 1 V 33 w 2 ( 2 u V 1 2 + 2 v V 1 V 2 + ( μ 2 - w 2 ) V 2 2 ) + p 11 ( μ 2 ( μ 2 - w 2 ) - D V 1 2 ) - 2 p 12 ( μ 2 v + D V 1 V 2 ) + p 22 ( 2 u μ 2 - D V 2 2 ) ] + ( 1 - τ V 1 ) 2 [ V 1 V 2 2 w 2 ( 2 u V 1 2 + 2 v V 1 V 2 + ( μ 2 + 2 w 2 ) V 2 2 ) + V 2 2 w 2 V 11 ( 2 u μ 2 - D V 2 2 - 2 u w 2 V 2 2 ) + 2 V 2 2 w 2 V 12 ( v μ 2 + D V 1 V 2 + 4 u w 2 V 1 V 2 + v w 2 V 2 2 ) + 2 V 2 2 ( - V 13 w ) ( μ 2 + 2 u V 1 2 - ( μ 2 - w 2 ) V 2 2 ) + V 2 2 V 22 ( 3 μ 2 - 8 u V 1 2 - 4 v V 1 V 2 - ( μ 2 - w 2 ) V 2 2 + μ 2 μ 2 w 2 - D w 2 V 1 2 ) - 4 V 1 2 V 2 ( - V 23 w ) ( 2 u V 1 + v V 2 ) + p 11 D V 1 3 + 2 p 12 D V 1 2 V 2 - 2 V 2 p 13 ( 2 v μ 2 + D V 1 V 2 ) + p 22 D V 1 V 2 2 + 2 V 2 p 23 ( 4 u μ 2 - D V 2 2 ) + μ 2 D p 123 ] + ( 1 - τ V 1 ) 3 [ - 3 V 1 2 V 2 4 + V 2 4 V 11 ( 2 u V 1 - 2 v V 2 + D w 2 V 1 ) + 2 V 2 3 V 12 ( - 2 μ 2 - 4 u V 1 2 + 2 v V 1 V 2 + ( μ 2 - w 2 ) V 2 2 + D w 2 V 1 2 ) + 2 V 1 V 2 3 ( - V 13 w ) ( V 1 v + V 2 ( μ 2 - w 2 ) ) + V 1 V 2 2 V 22 ( 8 u V 1 2 - ( μ 2 - w 2 ) V 2 2 + D w 2 V 1 2 ) - 2 V 1 2 V 2 2 ( - V 23 w ) ( V 1 v + V 2 ( μ 2 - w 2 ) ) + 2 V 2 2 p 13 ( - μ 2 ( μ 2 - w 2 ) + D V 1 2 ) + 2 V 2 2 p 23 ( - μ 2 v + D V 1 V 2 ) + V 2 2 p 33 ( 8 u μ 2 + μ 2 D w 2 - D V 2 2 ) ] + ( 1 - τ V 1 ) 4 [ V 1 3 V 2 4 + V 2 4 V 11 ( μ 2 + 2 v V 1 V 2 - ( μ 2 - w 2 ) V 2 2 ) - 6 V 1 2 V 2 4 V 12 v + V 1 2 V 2 3 V 22 ( 4 v V 1 + ( μ 2 - w 2 ) V 2 ) + V 2 3 p 33 ( 4 v μ 2 + D V 1 V 2 ) ] + ( 1 - τ V 1 ) 5 V 2 4 ( μ 2 - w 2 ) [ V 1 ( V 2 2 V 11 - 2 V 1 V 2 V 12 + V 1 2 V 22 ) + μ 2 p 33 ]
p 11 τ Δ ¯ Δ a 2 = p 11 a 2 + ( 1 - τ V 1 ) [ V 2 2 w 2 V 22 ( μ 2 - 2 u V 1 2 - 2 v V 1 V 2 - μ 2 V 2 2 ) + 2 V 1 V 2 3 ( - V 23 w ) - V 1 2 V 2 2 V 33 w 2 + V 1 p 11 ( 2 u V 1 2 - ( μ 2 - w 2 ) V 2 2 ) + 2 V 2 p 12 ( μ 2 - v V 1 V 2 - ( μ 2 - w 2 ) V 2 2 ) + 2 V 2 2 p 13 ( 2 u V 1 + v V 2 ) + p 123 ( 2 u μ 2 - D V 2 2 ) ] + ( 1 - τ V 1 ) 2 [ - V 1 2 V 2 4 w 2 - 2 V 2 3 w 2 V 12 ( μ 2 - v V 1 V 2 - μ 2 V 2 2 ) - V 1 V 2 2 w 2 V 22 ( 2 u V 1 2 - μ 2 V 2 2 ) - 2 V 1 V 2 4 ( - V 13 w ) + V 1 3 V 2 2 V 33 w 2 + V 1 2 V 2 p 11 ( 2 v V 1 + ( μ 2 - w 2 ) V 2 ) + 2 V 1 2 V 2 2 p 12 v + V 2 2 p 22 ( μ 2 - ( μ 2 - w 2 ) V 2 2 ) + 2 V 2 2 p 13 ( μ 2 - 2 u V 1 2 ) + 2 V 2 4 p 23 v + V 2 2 w 2 p 33 ( - 2 u w 2 V 2 2 + 2 u μ 2 - D V 2 2 ) + V 2 p 123 ( 2 v μ 2 + D V 1 V 2 ) ] + ( 1 - τ V 1 ) 3 [ V 1 3 V 2 4 w 2 + V 2 4 w 2 V 11 ( μ 2 - μ 2 V 2 2 ) - 2 v V 1 2 V 2 4 w 2 V 12 + V 1 2 V 2 3 w 2 V 22 ( 2 v V 1 + μ 2 V 2 ) + 2 V 1 2 V 2 4 ( - V 13 w ) - 2 V 1 3 V 2 3 ( - V 23 w ) + V 1 3 V 2 2 p 11 ( μ 2 - w 2 ) + 2 V 1 2 V 2 3 p 12 ( μ 2 - w 2 ) + V 1 V 2 4 p 22 ( μ 2 - w 2 ) - 2 V 1 2 V 2 3 p 13 v + 2 V 2 3 p 23 ( μ 2 - v V 1 V 2 ) + V 2 3 w 2 p 33 ( 2 u w 2 V 1 V 2 + 2 v μ 2 + D V 1 V 2 ) + V 2 2 p 123 μ 2 ( μ 2 - w 2 ) ] + ( 1 - τ V 1 ) 4 V 2 4 μ 2 w 2 [ V 1 ( V 2 2 V 11 - 2 V 1 V 2 V 12 + V 1 2 V 22 ) + μ 2 p 33 ] .
2 u V 1 2 = τ 2 ( 2 ū V 1 2 + 2 v ¯ V 1 V 2 + ( μ 2 - w 2 ) V 2 2 ) - 2 τ V 2 ( V 1 v ¯ + V 2 ( μ 2 - w 2 ) ) + V 2 2 ( μ 2 - w 2 ) v V 1 = τ ( V 1 v ¯ + V 2 ( μ 2 - w 2 ) ) - V 2 ( μ 2 - w 2 ) w = w ¯ D = τ 2 D ¯ .
x = - V 2 x y = - V 2 y z = - V 3 = 0 ,
x c = x a
V 2 = - a / c .
μ 2 - a 2 = 2 u V 1 2 + 2 v V 1 V 2 + ( μ 2 - w 2 ) V 2 2
2 u V 1 2 + 2 v V 1 V 2 + μ 2 V 2 2 = μ 2 V 3 = 0
V 2 = - a W 2 W 1 = - a c V 3 = W 3 - a c W 2 2 W 1 = 0 = W 3 - a W 2 ,
W 1 = c W 2 W 3 = a W 2 ,
W 1 = c f W 11 = c 2 f W 22 = f W 2 = f W 12 = c f W 23 = a f W 3 = a f W 13 = a c f + f W 33 = a 2 f .
q 11 = 0 q 22 = - f ( f + 2 a c f ) q 12 = a f f q 23 = c f f q 13 = - f f q 33 = 0 q 123 = - f 2 f .
γ 11 = c a ( a c f - f ) γ 22 = f γ 23 = a f γ 12 = c f γ 13 = a c f + f γ 33 = a c ( a c f - f )
Γ 11 = - a c f f Γ 22 = - 4 a c f f Γ 23 = 2 c f f Γ 12 = 2 a f f Γ 13 = - f f Γ 33 = - c a f f , Γ 123 = 0.
ω 11 = 1 a f [ μ 2 a c f - f ] ω 12 = 1 a θ f ω 13 = f a [ f ( 2 θ - μ 2 a c ) + f ] ω 22 = 1 a μ 2 c a f f ω 23 = 1 a f 2 f [ 2 μ 2 c a - θ ] ω 33 = 1 a f 3 [ f ( 4 μ 2 c a + μ 2 a c - 4 θ ) - f ]
π 11 = - 1 a ( a c f ) 2 f π 12 = 1 a 2 a c f f π 13 = - 1 a ( a c ) 2 f π 22 = - 1 a 4 ( a c ) 2 f π 23 = 1 a 2 ( a c ) 2 f f π 33 = - 1 a ( a c ) 2 f f 2 , π 123 = 0 ,
V 11 Δ = 1 a f [ μ 2 a c f - f ] V 12 Δ = 1 a [ f ( θ - μ 2 a c ) + f ] V 22 Δ = f a [ f ( μ 2 c a + μ 2 a c - 2 θ ) - f ] - V 13 c Δ = - V 23 c Δ = V 33 c 2 Δ = 0 ,
Δ = 1 a ( c a f ) 2 [ μ 2 c a f - f ( μ 2 μ 2 - θ 2 ) ] ,
p 11 Δ = 0 p 22 Δ = 0 p 12 Δ = 0 p 23 Δ = 0 p 13 Δ = 0 p 33 Δ = - 1 a ( a c ) 2 f f 2 , p 123 Δ = 0.
2 u = f 2 [ ( μ 2 c 2 / a 2 ) + μ 2 - ( 2 c θ / a ) ] v = f [ μ 2 - ( c θ / a ) ] w = c .
k 1 = 0 k 2 = ( μ 2 μ 2 - θ 2 ) { [ ( μ 2 c / a ) + ( μ 2 a / c ) - 2 θ ] f f - f 2 } μ 2 [ ( μ 2 c f / a ) - f ( μ 2 μ 2 - θ 2 ) ] .
x = a x / c
a / c = m 0 ,
( μ 2 m 0 + μ 2 m 0 - 2 θ ) f f = f 2 ,
f = C ( μ 2 m 0 + μ 2 m 0 - 2 θ ) 1 2 f = - C ( μ 2 m 0 + μ 2 m 0 - 2 θ ) 1 2 f = - C ( μ 2 m 0 + μ 2 m 0 - 2 θ ) 1 2 .
V 22 Δ = - C 2 a [ ( μ 2 / m 0 ) + μ 2 m 0 - 2 θ ] 2 [ μ 2 ( 1 m 0 - c a ) + μ 2 ( m 0 - a c ) ] .
2 u 0 = x 0 2 + y 0 2 = r 0 2 = 1 m 0 C 2
2 u 1 = x 1 2 + y 1 2 = r 1 2 = 1 m 1 C 2 = μ 2 m 0 m 2 C 2
V 1 = 0 = V 3 .
V 2 = ± μ / μ .
x = μ x / μ ,
V = ± μ v / μ .
p ¯ 11 0 ,
V 33 Δ ¯ = - a V 23 Δ ¯ 1 - τ V 1 τ = a 2 V 22 Δ ¯ ( 1 - τ V 1 τ ) 2 ,
V 22 Δ ¯ = 1 a c a τ [ ( a τ c ) 2 μ 2 f f - a τ c ( 2 θ f f + f 2 ) + μ 2 f f ] .
or             a τ / c = m 0 a τ / c = μ 2 / μ 2 m 0 ,
ξ = ξ x = x η = η y = y .
or             V 1 = V 3 = 0 ,             V 2 = - 1 V = - v .
τ = 1 ,             τ - 1 / V 1 = z / a .
( V 22 Δ ¯ / Δ ) a 2 = 0 ( V 23 Δ ¯ / Δ ) a 2 = 0 ( V 33 Δ ¯ / Δ ) a 2 = ( z / a ) ( μ 2 - μ 2 ) ,