## Abstract

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### Cited By

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### Figures (2)

Fig. 1

The curve of sine x, and the first-, third- and fifth-order approximations.

Fig. 2

The curve of tangent x, and the first-, third- and fifth-order approximations.

### Equations (79)

$sin u = u - u 3 / 6 + u 5 / 120 , tan u = u + u 3 / 3 + 2 u 5 / 15.$
$u 1 = 1 2 ( x ¯ 1 2 + x ¯ 2 2 ) , u 2 = ( x ¯ 1 x ¯ 3 + x ¯ 2 x ¯ 4 ) , u 3 = ( x ¯ 3 2 + x ¯ 4 2 ) .$
$A i k u i u k = A 11 u 1 2 + 2 A 12 u 1 u 2 + 2 A 13 u 1 u 3 + A 22 u 2 2 + 2 A 23 u 2 u 3 + A 33 u 3 2 ,$
$E = E 0 + Ā i ū i + 1 2 Ā i k ū i ū k + 1 6 A i k λ u i u k u λ .$
$ξ 1 = - ∂ E / ∂ x ¯ 1 , ξ ′ 3 = ∂ E / ∂ x ¯ ′ 3 , ξ 2 = - ∂ E / ∂ x ¯ 2 , ξ ′ 4 = ∂ E / ∂ x ¯ ′ 4 ,$
$- ξ 1 = ∂ E ∂ u ˜ 1 x ¯ 1 + ∂ E ∂ ū 2 x ¯ ′ 3 = ( Ā 1 + Ā 1 i ū i + 1 2 Ā 1 i k ū i ū k ) x ¯ 1 + ( Ā 2 + Ā 2 i ū i + 1 2 Ā 2 i k ū i ū k ) x ¯ ′ 3 , ξ ′ 3 = ∂ E ∂ ū 2 x ¯ 1 + ∂ E ∂ ū 3 x ¯ ′ 3 = ( Ā 2 + Ā 2 i ū i + 1 2 Ā 2 i k ū i ū k ) x ¯ 1 + ( Ā 3 + Ā 3 i ū i + 1 2 Ā 3 i k ū i ū k ) x ¯ ′ 3 .$
$x i = Ā 2 x ¯ i , u i = Ā 2 2 ū i$
$- ξ 1 = ( A 1 + A 1 i u i + 1 2 A 1 i k u i u k ) x 1 + ( 1 + A 2 i u i + 1 2 A 2 i k u i u k ) x ′ 3 , ξ 3 = ( 1 + A 2 i u i + 1 2 A 2 i k u i u k ) x 1 + ( A 3 + A 3 i u i + 1 2 A 3 i k u i u k ) x ′ 3 ,$
$A i = Ā i / Ā 2 , A i k = Ā i k / Ā 2 2 , A i k λ = Ā i k λ / Ā 2 3 .$
$sin σ = ( ξ 2 1 + ξ 2 2 ) 1 2 , tan σ = ( ξ 2 1 + ξ 2 2 ) 1 2 ( 1 - ( ξ 2 1 + ξ 2 2 ) ) 1 2 , sin σ ′ = ( ξ ′ 2 3 + ξ ′ 2 4 ) 1 2 , tan σ ′ = ( ξ ′ 2 3 + ξ ′ 2 4 ) 1 2 ( 1 - ( ξ ′ 2 3 + ξ ′ 2 4 ) ) 1 2 .$
$— Ξ 1 = ξ 1 ( 1 - ( ξ ′ 2 3 + ξ ′ 2 4 ) ) 1 2 = ( B 1 + B 1 i u i + 1 2 B 1 i k u i u k ) x 1 + ( 1 + B 2 i u i + 1 2 B 2 i k u i u k ) x ′ 3 , Ξ 3 = ξ ′ 3 ( 1 - ( ξ ′ 2 3 + ξ ′ 2 4 ) ) 1 2 = ( 1 + B ′ 2 i u i + 1 2 B ′ 2 i k u i u k ) x 1 + ( B ′ 3 + B ′ 3 i u i + 1 2 B ′ 3 i k u i u k ) x ′ 3 ,$
$x ′ 1 g = x ′ 3 + λ ξ ′ 3 , x ′ 2 g = x ′ 4 + λ ξ ′ 4 , or x ′ 1 g = x ′ 3 + g ′ Ξ 3 , x ′ 2 g = x ′ 4 + g ′ Ξ 4 , g ′ = λ ( 1 - ( ξ ′ 2 3 + ξ ′ 2 4 ) ) 1 2 .$
$x ′ 1 g = g ( 1 + B ′ 2 i u i + 1 2 B ′ 2 i k u i u k ) x 1 + g ′ ( 1 / g ′ + B ′ 3 + B ′ 3 i u i + 1 2 B ′ 3 i k u i u k ) x ′ 3 .$
$- B ′ 3 x ′ - x 1 = ( B ′ 2 i u i + 1 2 B ′ 2 i k u i u k ) x 1 + ( B ′ 3 i u i + 1 2 B ′ 3 i k u i u k ) x ′ 3 .$
$x ′ = 1 B ′ 3 x 1 = β ′ x 1 ,$
$x 3 g = x 1 + g ′ Ξ 1 = - g ′ [ B 1 - 1 / g ′ + B 1 i u i + 1 2 B 1 i k u i u k ] x 1 - g ′ [ 1 + B 2 i u i + 1 2 B 2 i k u i u k ] x ′ 3 .$
$- ( B 1 x 3 * + x ′ 3 ) = [ B 1 i u i + 1 2 B 1 i k u i u k ] x 1 + [ B 2 i u i + 1 2 B 2 i k u i u k ] x ′ 3 .$
$x ′ 3 = - B 1 x 3 * = B ′ P x 3 * ,$
$Ξ 1 = ξ 1 ( 1 - ( ξ 2 1 + ξ 2 2 ) ) 1 2 = ξ 1 [ 1 + 1 2 ( ξ 2 1 + ξ 2 2 ) + 3 8 ( ξ 2 1 + ξ 2 2 ) 2 ] , Ξ ′ 3 = ξ ′ 3 ( 1 - ( ξ ′ 2 3 + ξ ′ 2 4 ) ) 1 2 = ξ ′ 3 [ 1 + 1 2 ( ξ ′ 2 3 + ξ ′ 2 4 ) + 3 8 ( ξ ′ 2 3 + ξ ′ 2 4 ) 2 ] .$
$1 2 ( ξ 2 1 + ξ 2 2 ) = A 1 2 u 1 + A 1 u 2 + u 3 + 2 A 1 A 11 u 1 2 + ( 3 A 1 A 12 + A 11 ) u 1 u 2 + 2 ( A 1 A 13 + A 12 ) u 1 u 3 + ( A 1 A 22 + A 12 ) u 2 2 + ( A 1 A 23 + 2 A 22 + A 13 ) u 2 u 3 + 2 A 23 u 3 2$
$B 1 = A 1 , B ′ 3 = A 3 ,$
$B 11 = A 11 + A 1 3 , B 12 = A 12 + A 1 2 , B ′ 21 = A 12 + 1 , B 13 = A 13 + A 1 , B ′ 22 = A 22 + A ′ 3 , B 21 = A 12 + A 1 2 , B ′ 23 = A 23 + A ′ 2 3 , B 22 = A 22 + A 1 , B ′ 31 = A 13 + A ′ 3 , B 23 = A 23 + 1 , B ′ 32 = A 23 + A ′ 2 3 , B ′ 33 = A 33 + A ′ 3 3 ,$
$B 111 = A 111 + 6 A 1 2 A 11 + 3 A 1 5 , B ′ 211 = A 112 + 6 A 12 + 3 , B 112 = A 112 + 4 A 1 2 A 12 + 2 A 1 A 11 + 3 A 1 4 , B ′ 212 = A 122 + 2 A 3 A 12 + 3 A 22 + A 13 + 3 A 3 , B 113 = A 113 + 3 A 1 2 A 13 + 2 A 1 A 12 + A 11 + 3 A 1 3 , B ′ 213 = A 123 + A 3 2 A 12 + 2 A 3 A 13 + 3 A 23 + 3 A 3 2 , B 122 = A 122 + 2 A 1 2 A 22 + 4 A 1 A 12 + 3 A 1 3 , B ′ 222 = A 222 + 4 A 3 A 22 + 2 A 23 + 3 A 3 2 , B 123 = A 123 + A 1 2 A 23 + 2 A 1 ( A 13 + A 22 ) + A 12 + 3 A 1 2 , B ′ 223 = A 223 + A 3 2 A 22 + 4 A 3 A 23 + A 33 + 3 A 3 3 , B 133 = A 133 + 4 A 1 A 23 + 2 A 13 + 3 A 1 , B ′ 233 = A 233 + 2 A 3 2 A 23 + 4 A 3 A 33 + 3 A 3 4 , B 211 = A 112 + 2 A 1 2 A 12 + 4 A 1 A 11 + 3 A 1 4 , B ′ 311 = A 113 + 4 A 3 A 12 + 2 A 13 + 3 A 3 , B 212 = A 122 + A 1 2 A 22 + 4 A 1 A 12 + A 11 + 3 A 1 3 , B ′ 312 = A 123 + A 3 2 A 12 + 2 A 3 ( A 13 + A 22 ) + A 23 + 3 A 3 2 , B 213 = A 123 + A 1 2 A 23 + 2 A 1 A 13 + 3 A 12 + 3 A 1 2 , B ′ 313 = A 133 + 3 A 3 2 A 13 + 2 A 3 A 23 + A 33 + 3 A 3 3 , B 222 = A 222 + 4 A 1 A 22 + 2 A 12 + 3 A 1 2 , B ′ 322 = A 223 + 2 A 3 2 A 22 + 4 A 3 A 23 + 3 A 3 3 , B 223 = A 223 + 2 A 1 A 23 + ( 3 A 22 + A 13 ) + 3 A 1 , B ′ 323 = A 233 + 4 A 3 2 A 23 + 2 A 3 A 33 + 3 A 3 4 , B 233 = A 233 + 6 A 23 + 3 , B ′ 333 = A 333 + 6 A 3 2 A 33 + 3 A 3 5 .$
$A i κ λ = A κ i λ , B i κ λ not necessarily = B κ i λ .$
$B 11 = A 11 + B 1 3 , B 12 = A 12 + B 1 2 = B 21 , B ′ 21 = A 12 + 1 , B 13 = A 13 + B 1 , B ′ 31 = A 13 + B ′ 3 , B 22 = A 22 + B 1 , B ′ 22 = A 22 + B ′ 3 , B 23 = A 23 + 1 , B ′ 23 = A 23 + B ′ 2 3 = B ′ 32 , B ′ 33 = A 33 + B ′ 3 3 .$
$B ′ 23 - B ′ 32 = 0 , B 21 - B 12 = 0.$
$B ′ 23 - B 23 = B ′ 2 3 - 1 = α 1 , B ′ 22 - B 22 = B ′ 13 - B 13 = B ′ 3 - B 1 = α 2 , B ′ 12 - B 12 = 1 - B 1 2 = α 3 .$
$C 111 = A 111 - 3 B 1 5 , C 112 = A 112 - 3 B 1 4 , C ′ 112 = A 112 - 3 , C 113 = A 113 - 3 B 1 3 , C 122 = A 122 - 3 B 1 3 , C ′ 122 = A 122 - 3 B 3 , C ′ 113 = A 113 - 3 B 3 , C 123 = A 123 - 3 B 1 2 , C 222 = A 222 - 3 B 1 2 , C ′ 222 = A 222 - 3 B 3 2 , C ′ 123 = A 123 - 3 B 3 2 , C 133 = A 133 - 3 B 1 , C 223 = A 223 - 3 B 1 , C ′ 223 = A 223 - 3 B 3 3 , C ′ 133 = A 133 - 3 B 3 3 , C 233 = A 233 - 3 , C ′ 233 = A 233 - 3 B 3 4 , C ′ 333 = A 333 - 3 B 3 5 .$
$B 111 = C 111 + 6 B 1 2 B 11 , B 112 = C 112 + 4 B 1 2 B 12 + 2 B 1 B 11 , B 211 = C 112 + 2 B 1 2 B 12 + 4 B 1 B 11 , B 113 = C 113 + 3 B 1 2 B 13 + 2 B 1 B 12 + B 11 , B 122 = C 122 + 2 B 1 2 B 22 + 4 B 1 B 12 , B 212 = C 122 + B 1 2 B 22 + 4 B 1 B 12 + B 11 , B 123 = C 123 + B 1 2 B 23 + 2 B 1 ( B 13 + B 22 ) + B 12 , B 213 = C 123 + B 1 2 B 23 + 2 B 1 B 13 + 3 B 12 , B 222 = C 222 + 4 B 1 B 22 + 2 B 12 , B 133 = C 133 + 4 B 1 B 23 + 2 B 13 , B 223 = C 223 + 2 B 1 B 23 + ( 3 B 22 + B 13 ) , B 233 = C 233 + 6 B 23 ,$
$B ′ 211 = C ′ 112 + 6 B ′ 12 , B ′ 212 = C ′ 122 + 2 B ′ 3 B ′ 12 + 3 B ′ 22 + B ′ 13 , B ′ 311 = C ′ 113 + 4 B ′ 3 B ′ 12 + 2 B ′ 13 , B ′ 222 = C ′ 222 + 4 B ′ 3 B ′ 22 + 2 B ′ 23 , B ′ 213 = C ′ 123 + B ′ 2 3 B ′ 12 + 2 B ′ 3 B ′ 13 + 3 B ′ 23 , B ′ 312 = C ′ 123 + B ′ 2 3 B ′ 12 + 2 B ′ 3 ( B ′ 13 + B ′ 22 ) + B ′ 23 , B ′ 223 = C ′ 223 + B ′ 2 3 B ′ 22 + 4 B ′ 3 B ′ 23 + B ′ 33 , B ′ 322 = C ′ 223 + 2 B ′ 2 3 B ′ 22 + 4 B ′ 3 B ′ 23 , B ′ 313 = C ′ 133 + 3 B ′ 2 3 B ′ 13 + 2 B ′ 3 B ′ 23 + B ′ 33 , B ′ 233 = C ′ 233 + 2 B ′ 2 3 B ′ 23 + 4 B ′ 3 B ′ 33 , B ′ 323 = C ′ 233 + 4 B ′ 2 3 B ′ 23 + 2 B ′ 3 B ′ 33 , B ′ 333 = C ′ 333 + 6 B ′ 2 3 B ′ 33 .$
$B 311 - B 112 = 2 B 1 ( B 11 - B 1 B 12 ) , B ′ 312 - B ′ 213 = 2 ( B ′ 3 B ′ 22 - B ′ 23 ) , B 212 - B 122 = B 11 - B 1 2 B 22 , B ′ 322 - B ′ 223 = B ′ 2 3 B ′ 22 - B ′ 33 , B 213 - B 123 = 2 ( B 12 - B 1 B 22 ) , B ′ 323 - B ′ 233 = 2 B ′ 3 ( B ′ 3 B ′ 23 - B ′ 33 ) ,$
$C ′ 112 - C 112 = 3 ( B 1 4 - 1 ) = γ 5 , C ′ 122 - C 122 = C ′ 113 - C 113 = 3 ( B 1 3 - B ′ 3 ) = γ 4 , C ′ 123 - C 123 = C ′ 222 - C 222 = 3 ( B 1 2 - B ′ 2 3 ) = γ 3 , C ′ 223 - C 223 = C ′ 133 - C 133 = 3 ( B 1 - B ′ 3 3 ) = γ 2 , C ′ 233 - C 233 = 3 ( 1 - B ′ 4 3 ) = γ 1 .$
$z c = 1 2 D 1 ( x c 2 + y c 2 ) + 1 8 D 2 3 ( x c 2 + y c 2 ) 2 = D 1 u c + 1 2 D 2 3 u c 2 , z ′ c = D 3 u ′ c + 1 2 D 4 3 u ′ 2 c .$
$z ¯ c = ( D 1 / Ā 2 ) ū c + 1 2 ( D 2 / Ā 2 ) 3 ū c 2 , z ¯ ′ c = ( D 3 / Ā 2 ) ū ′ c + 1 2 ( D 4 / Ā 2 ) 3 ū ′ 2 c .$
$D 1 = D 2 ( D 3 = D 4 ) ,$
$An ellipsoid for ( D 1 / D 2 ) > 0 ( ( D 3 / D 4 ) > 0 ) , A hyperboloid for ( D 1 / D 2 ) < 0 ( ( D 3 / D 4 ) < 0 ) ,$
$a = D 1 2 / D 2 3 , b = ( ( - ) D 1 / D 2 3 ) 1 2 ,$
$- B ′ 3 x ′ - x 1 = [ B ′ 2 i u i + 1 2 B ′ 2 i κ u i u κ ] x 1 + [ B ′ 3 i u i + 1 2 B ′ 3 i κ u i u κ ] x ′ 3 ,$
$- B ′ 3 x ′ c - x c = [ B ′ 2 i c u i c + 1 2 B ′ 2 i κ c u i c u κ c ] x c + [ B ′ 3 i c u i c + 1 2 B ′ 3 i κ c u i c u κ c ] x ′ 3 c .$
$x c = x 1 + z c Ξ 1 , x ′ c = x ′ + z ′ c Ξ 3 .$
$x c = x 1 - ( D 1 B 1 u 1 x 1 + D 1 u 1 x ′ 3 ) - 1 2 [ ( 2 D 1 B 11 - 4 D 1 2 B 1 2 + D 2 3 B 1 ) u 1 2 + 2 ( D 1 B 12 - D 1 2 B 1 ) u 1 u 2 + 2 D 1 B 13 u 1 u 3 ] x 1 - 1 2 [ ( 2 D 1 B 12 - 4 D 1 2 B 1 + D 2 3 ) u 1 2 + 2 ( D 1 B 22 - D 1 2 ) u 1 u 2 + 2 D 1 B 23 u 1 u 3 ] x ′ 3 ,$
$u 1 c = u 1 - 2 B 1 D 1 u 1 2 - D 1 u 1 u 2 , u 2 c = u 2 - D 1 B 1 u 1 u 2 - 2 D 1 u 1 u 3 , u 3 c = u 3 .$
$- B ′ 3 x ′ c - x c = [ ( B ′ 21 - D 3 B ′ 3 ) u 1 + B ′ 22 u 2 + B ′ 23 u 3 ] x 1 + [ ( B ′ 13 - D 3 ) u 1 + B ′ 23 u 2 + B ′ 33 u 3 ] x ′ 3 + { 1 2 [ B ′ 212 - D 3 B ′ 3 ( 6 B ′ 12 - 4 D 3 B ′ 3 ) - D 4 B ′ 2 3 ] u 1 2 + [ B ′ 212 - D 3 B ′ 3 ( 3 B ′ 22 + B ′ 13 - D 3 ) ] u 1 u 2 + [ B ′ 213 - 3 D 3 B ′ 3 B ′ 23 ] u 1 u 3 + 1 2 [ B ′ 222 - 2 D 3 B ′ 3 B ′ 23 ] u 2 2 + [ B ′ 223 - D 3 B ′ 3 B ′ 33 ] u 2 u 3 + 1 2 B ′ 233 u 3 2 } x 1 + { 1 2 [ B ′ 311 - D 3 B ′ 3 ( 2 B ′ 13 + 4 B ′ 3 B ′ 12 - 4 D 3 ) ] u 1 2 + [ B ′ 312 - D 3 B ′ 3 ( B ′ 23 + B ′ 3 ( 2 B ′ 22 + B ′ 13 ) - D 3 ) ] u 1 u 2 + [ B ′ 313 - D 3 B ′ 3 ( B ′ 33 + 2 B ′ 3 B ′ 23 ) ] u 1 u 3 + 1 2 [ B ′ 322 - D 3 B ′ 23 ] u 2 2 + [ B ′ 323 - D 3 B ′ 33 ] u 2 u 3 + 1 2 B ′ 333 u 3 2 ] } x ′ 3 .$
$B ′ 12 c = B ′ 12 + D 1 B 1 - D 3 / B ′ 3 , B ′ 13 c = B ′ 13 + D 1 - D 3 , B ′ 22 c = B ′ 22 , B ′ 23 c = B ′ 23 , B ′ 33 c = B ′ 33 ,$
$B ′ 211 c = B ′ 211 + 2 D 1 B 4 + 6 B ′ 12 ( B 1 D 1 - D 3 B ′ 3 ) + ( B 1 D 2 3 - D 4 3 B ′ 3 3 ) + 2 ( B 1 D 1 - D 3 B ′ 3 ) ( B 1 D 1 - 2 D 3 B ′ 3 ) , B ′ 212 c = B ′ 212 + D 1 B 12 + D 1 ( B ′ 12 + 2 B 1 B ′ 22 ) - D 3 B ′ 3 ( B ′ 13 + 3 B ′ 22 ) + D 3 B ′ 3 ( D 3 - D 1 ) , B ′ 311 c = B ′ 311 + 2 D 1 B 12 + 2 D 1 ( B ′ 12 + 2 B 1 B ′ 13 ) - 2 D 3 B ′ 3 ( B ′ 13 + 2 B ′ 3 B ′ 12 ) + ( D 2 3 - D 4 3 B ′ 2 3 ) + 2 ( D 3 B ′ 3 - B 1 D 1 ) ( 2 D 3 - D 1 ) , B ′ 213 c = B ′ 213 + D 1 B 13 + D 1 ( 2 B ′ 22 + B 1 B ′ 23 ) - 3 D 3 B ′ 3 / B ′ 3 , B ′ 222 c = B ′ 222 - 2 D 3 B ′ 23 / B ′ 3 , B ′ 312 c = B ′ 312 + D 1 B 22 + D 1 ( B ′ 13 + B ′ 22 + B 1 B ′ 23 ) - ( D 3 / B ′ 3 ) ( B ′ 23 + B ′ 3 ( B ′ 13 + 2 B ′ 22 ) ) + D 3 ( D 3 - D 1 ) , B ′ 223 c = B ′ 223 - D 3 B ′ 33 / B ′ 3 , B ′ 313 c = B ′ 313 + D 1 B 23 + 3 D 1 B ′ 23 - ( D 3 / B ′ 3 ) ( B ′ 33 + 2 B ′ 3 B ′ 23 ) , B ′ 322 c = B ′ 322 - 2 D 3 B ′ 23 , B ′ 233 c = B ′ 233 , B ′ 323 c = B ′ 323 - D 3 B ′ 33 , B ′ 333 c = B ′ 333 .$
$z * = 1 2 E 3 ( x * 3 2 + y * 3 2 ) = E 3 u * 3 .$
$x * 3 = x ′ 3 + E 3 u 3 Ξ 3 = E 3 u 3 x 1 + ( 1 + E 3 B ′ 3 u 3 ) x ′ 3 ) , u * 1 = u 1 , u * 2 = 2 E 3 u 3 u 1 + ( 1 + E 3 B ′ 3 u 3 ) u 2 , u * 3 = u 3 + E 3 u 2 u 3 + 2 E 3 B ′ 3 u 3 2 .$
$- B ′ 3 x ′ - x 1 = [ B * ′ 2 i u * i + 1 2 B * ′ 2 i κ u * i u * κ ] x 1 + [ B * ′ 3 i u * i + 1 2 B * ′ 3 i κ u * i u * κ ] x * 3 ,$
$B * ′ 12 = B ′ 12 , B * ′ 23 = B ′ 23 , B * ′ 22 = B ′ 22 , B * ′ 33 = B ′ 33 . B * ′ 13 = B ′ 13 ,$
$B * ′ 211 = B ′ 211 , B * ′ 212 = B ′ 212 , B * ′ 311 = B ′ 311 , B * ′ 213 = B ′ 213 - E 3 ( 2 B ′ 22 + B ′ 13 ) , B * ′ 222 = B ′ 222 , B * ′ 312 = B ′ 312 , B ′ 223 = B ′ 223 - E 3 ( 2 B ′ 23 + B ′ 3 B ′ 22 ) , B * ′ 313 = B ′ 313 - E 3 ( 2 B ′ 23 + B ′ 3 B ′ 13 ) , B * ′ 322 = B ′ 322 , B * ′ 233 = B ′ 233 - 2 E 3 ( B ′ 33 + 2 B ′ 3 B ′ 23 ) , B * ′ 323 = B ′ 323 - E 3 ( B ′ 33 + 2 B ′ 3 B ′ 23 ) , B * ′ 333 = B ′ 333 - 6 E 3 B ′ 3 B ′ 33 .$
$x ¯ 1 = x ¯ ˙ 1 , x ¯ 3 = x ¯ ˙ 3 + k ′ Ξ ′ 3$
$x 3 / A 2 = x 3 / A 2 + k ′ Ξ ′ 3 , x 1 / A 1 = x 1 / A 2 .$
$κ 3 = A 2 / A 2 , λ 3 = k ′ A 2 ,$
$x 3 = κ 3 x 3 + λ 3 Ξ ′ 3 = κ 3 x 3 + λ 3 ( 1 + B ′ 2 i u i + 1 2 B ′ 2 i κ u i u κ ) x 1 + λ 3 ( B ′ 3 + B ′ 3 i u i + 1 2 B ′ 3 i κ u i u κ ) x 3 = λ 3 x 1 + ( κ 3 + B ′ 3 λ 3 ) x 3 + λ 3 ( B ′ 2 i u i + 1 2 B ′ 2 i κ u i u κ ) x 1 + λ 3 ( B ′ 3 + B ′ 3 i u i + 1 2 B ′ 3 i κ u i u κ ) x 3 x 1 = κ 3 x 1 .$
$x 3 = λ 3 x 1 + ( κ 3 + B ′ 3 λ 3 ) x 3 , x 1 = κ 3 x 13 ,$
$Ξ 3 = x 1 + B ′ 3 x 3 = x 1 + B ′ 3 x 3 .$
$B ′ 3 = B ′ 3 , κ 3 + B ′ 3 λ 3 = 1.$
$- Ξ 1 = B 1 x 1 + x ′ 3 = B 1 x 1 + x ′ 3 ,$
$B 1 κ 3 + λ 3 = B 1 .$
$κ 3 = 1 - B 1 B ′ 3 1 - B 1 B ′ 3 , λ 3 = B 1 - B 1 1 - B 1 B ′ 3 .$
$x 1 = κ 3 x 1 , x 3 = λ 3 x 1 + x 3 + λ 3 ( B ′ 2 i u i + 1 2 B ′ 2 i κ u i u κ ) x 1 + λ 3 ( B ′ 3 i u i + 1 2 B ′ 3 i κ u i u κ ) x ′ 3$
$u 1 = κ 3 2 u 1 , u 2 = 2 κ 3 λ 3 u 1 + κ 3 u 2 + κ 3 λ 3 ( 2 B ′ 2 i u i u 1 + B ′ 3 i u i u 2 ) , u 3 = λ 3 2 u 1 + λ 3 u 2 + u 3 + λ 3 [ B ′ 2 i u i ( 2 λ 3 u 1 + u 2 ) + B ′ 3 i u i ( λ 3 u 2 + 2 u 3 ) ] .$
$Ξ ′ 3 = [ 1 + B ′ 2 i u i + 1 2 B ′ 2 i κ u i u κ ] x 1 + [ B ′ 3 + B ′ 3 i u i + 1 2 B ′ 3 i κ u i u κ ] x ′ 3 = [ 1 + B ′ 2 i u i + 1 2 B ′ 2 i κ u i u κ ] x 1 + [ B ′ 3 + B ′ 3 i u i + 1 2 B ′ 3 i κ u i u κ ] x ′ 3$
$κ 3 B ′ 12 = κ 3 3 B ′ 12 + κ 3 2 λ 3 ( 2 B ′ 22 + B ′ 13 ) + 3 κ 3 λ 3 2 B ′ 23 + λ 3 3 B ′ 33 , κ 3 B ′ 13 = κ 3 2 B ′ 13 + 2 κ 3 λ 3 B ′ 23 + λ 3 2 B ′ 33 , κ 3 B ′ 22 = κ 3 2 B ′ 22 + 2 κ 3 λ 3 B ′ 23 + λ 3 2 B ′ 33 , κ 3 B ′ 23 = κ 3 B ′ 23 + λ 3 B ′ 33 , κ 3 B ′ 33 = B ′ 33 ,$
$κ 3 B ′ 211 = κ 3 5 B ′ 211 + κ 3 4 λ 3 ( B ′ 311 + 4 B ′ 212 ) + 2 κ 3 3 λ 3 2 [ B ′ 213 + 2 ( B ′ 222 + B ′ 312 ) ] + 2 κ 3 2 λ 3 3 [ 2 ( B ′ 223 + B ′ 322 ) + B ′ 313 ] + κ 3 λ 3 4 ( B ′ 233 + 4 B ′ 323 ) + λ 3 5 B ′ 333 + 2 κ 3 λ 3 B ′ 12 ( B ′ 13 + 2 B ′ 22 ) , κ 3 B ′ 212 = κ 3 4 B ′ 212 + κ 3 3 λ 3 ( B ′ 213 + 2 B ′ 222 + B ′ 312 ) + κ 3 2 λ 3 2 ( 3 B ′ 223 + B ′ 313 + 2 B ′ 322 ) + κ 3 λ 3 3 ( B ′ 233 + 3 B ′ 323 ) + λ 3 4 B ′ 333 + 2 κ 3 λ 3 [ B ′ 12 B ′ 23 + B ′ 22 ( B ′ 13 + B ′ 22 ) ] , κ 3 B ′ 311 = κ 5 4 B ′ 311 + 4 κ 3 3 λ 3 B ′ 312 + 2 κ 3 2 λ 3 2 ( B ′ 313 + 2 B ′ 322 ) + 4 κ 3 λ 3 3 B ′ 323 + λ 3 4 B ′ 333 + 2 κ 3 λ 3 ( B ′ 2 13 + 2 B ′ 12 B ′ 23 ) , κ 3 B ′ 213 = κ 3 3 B ′ 213 + κ 3 2 λ 3 ( 2 B ′ 223 + B ′ 313 ) + κ 3 λ 3 2 ( B ′ 233 + 2 B ′ 323 ) + λ 3 3 B ′ 333 + κ 3 λ 3 [ B ′ 12 B ′ 33 + B ′ 22 ( 3 B ′ 13 + 2 B ′ 22 ) ] , κ 3 B ′ 222 = κ 3 3 B ′ 222 + κ 3 2 λ 3 ( 2 B ′ 223 + B ′ 322 ) + κ 3 λ 3 2 ( B ′ 233 + 2 B ′ 323 ) + λ 3 3 B ′ 333 + 6 κ 3 λ 3 B ′ 22 B ′ 23 , κ 3 B ′ 312 = κ 3 3 B ′ 312 + κ 3 2 λ 3 ( B ′ 313 + 2 B ′ 322 ) + 3 κ 3 λ 3 2 B ′ 323 + λ 3 3 B ′ 333 + κ 3 λ 3 [ B ′ 12 B ′ 33 + B ′ 23 ( 2 B ′ 22 + 3 B ′ 13 ) ] , κ 3 B ′ 223 = κ 3 2 B ′ 223 + κ 3 λ 3 ( B ′ 233 + B ′ 323 ) + λ 3 2 B ′ 333 + 2 κ 3 λ 3 [ 2 B ′ 2 23 + B ′ 22 B ′ 33 ] , κ 3 B ′ 313 = κ 3 2 B ′ 313 + 2 κ 3 λ 3 B ′ 323 + λ 3 2 B ′ 333 + 2 κ 5 λ 3 ( B ′ 2 23 + 2 B ′ 13 B ′ 33 ) , κ 3 B ′ 322 = κ 3 2 B ′ 322 + 2 κ 3 λ 3 B ′ 323 + λ 3 2 B ′ 333 + 2 κ 3 λ 3 ( 2 B ′ 2 23 + B ′ 22 B ′ 33 ) , κ 3 B ′ 233 = κ 3 B ′ 233 + λ 3 B ′ 333 + 6 κ 3 λ 3 B ′ 23 B ′ 33 , κ 3 B ′ 323 = κ 3 B ′ 323 + λ 3 B ′ 333 + 6 κ 3 λ 3 B ′ 23 B ′ 33 , κ 3 B ′ 333 = B ′ 333 + 6 κ 3 λ 3 B ′ 2 33 .$
$κ 3 C ′ 112 = κ 3 5 C ′ 112 + κ 3 4 λ 3 ( 3 C ′ 122 + C ′ 113 ) + 2 κ 3 3 λ 3 2 ( 3 C ′ 123 + 2 C ′ 222 ) + 2 κ 3 2 λ 3 3 ( C ′ 133 + 4 C ′ 223 ) + 5 κ 3 λ 3 4 C ′ 233 + λ 3 5 C ′ 333 + 2 κ 3 λ 3 B ′ 12 ( B ′ 13 + 2 B ′ 22 ) , κ 3 C ′ 122 = κ 3 4 C ′ 122 + 2 κ 3 3 λ 3 ( C ′ 123 + C ′ 222 ) + κ 3 2 λ 3 2 ( C ′ 133 + 5 C ′ 223 ) + 4 κ 3 λ 3 3 C ′ 223 + λ 3 4 C ′ 333 + 2 κ 3 λ 3 [ B ′ 12 B ′ 23 + B ′ 22 ( B ′ 13 + B ′ 22 ) ] , κ 3 C ′ 113 = κ 3 4 C ′ 113 + 4 κ 3 3 λ 3 C ′ 123 + 2 κ 3 2 λ 3 2 ( C ′ 133 + 2 C ′ 223 ) + 4 κ 3 λ 3 2 C ′ 233 + λ 3 4 C ′ 333 + 2 κ 3 λ 3 [ B ′ 2 13 + 2 B ′ 12 B ′ 23 ] , κ 3 C ′ 123 = κ 3 3 C ′ 123 + κ 3 2 λ 3 ( 2 C ′ 223 + C ′ 133 ) + 3 κ 3 λ 3 2 C ′ 233 + λ 3 3 C ′ 333 + κ 3 λ 3 [ B ′ 12 B ′ 33 + B ′ 23 ( 3 B ′ 13 + 2 B ′ 22 ) ] , κ 3 C ′ 222 = κ 3 3 C ′ 222 + 3 κ 3 2 λ 3 C ′ 223 + 3 κ 3 λ 3 2 C ′ 233 + λ 3 3 C ′ 333 + 6 κ 3 λ 3 B ′ 22 B ′ 23 , κ 3 C ′ 133 = κ 3 2 C ′ 133 + 2 κ 3 λ 3 C ′ 233 + λ 3 2 C ′ 333 + 2 κ 3 λ 3 [ B ′ 2 23 + 2 B ′ 13 B ′ 33 ] , κ 3 C ′ 223 = κ 3 2 C ′ 223 + 2 κ 3 λ 3 C ′ 233 + λ 3 2 C ′ 333 + 2 κ 3 λ 3 [ 2 B ′ 2 23 + B ′ 22 B ′ 33 ] , κ 3 C ′ 233 = κ 3 C ′ 233 + λ 3 C ′ 333 + 6 κ 3 λ 3 B ′ 23 B ′ 33 , κ 3 C ′ 333 = C ′ 333 + 6 κ 3 λ 3 B ′ 2 33 .$
$x 1 = κ 1 x 1 - λ 1 Ξ 1 , x ′ 3 = κ 1 x 3 .$
$- Ξ 1 = B 1 x 1 + x ′ 3 = B 1 x 1 + x 3 , Ξ 3 = x 1 + B ′ 3 x ′ 3 = x 1 + B 3 x 3 ,$
$x 1 = ( κ 1 + λ 1 B 1 ) x 1 + λ 1 x 3 , x 3 κ 1 x 3 .$
$B 1 = B 1 , κ 1 + λ 1 B 1 = 1 , λ 1 + κ 1 B ′ 3 = B 3 ,$
$κ 1 = 1 - B 1 B 3 1 - B 1 B ′ 3 , λ 1 = B 3 - B ′ 3 1 - B 1 B ′ 3 .$
$x 1 = x 1 + λ 1 x 3 + λ 1 ( B 1 i u i + 1 2 B 1 i κ u i u κ ) x 1 + λ 1 ( B 2 i u i + 1 2 B 2 i κ u i u κ ) x 3$
$x ′ 3 = κ 1 x 3 .$
$- Ξ 1 = ( B 1 + B 1 i u i + 1 2 B 1 i κ u i u κ ) x 1 + ( 1 + B 2 i u i + 1 2 B 2 i κ u i u κ ) x ′ 3 = ( B 1 + B 1 i u i + 1 2 B 1 i κ u i u κ ) x 1 + ( 1 + B 2 i u i + 1 2 B 2 i κ u i u κ ) x 3 ,$
$κ 1 B ′ 12 = κ 1 B ′ 12 + λ 1 B 11 , κ 1 B ′ 13 = κ 1 2 B ′ 13 + κ 1 λ 1 ( 2 B ′ 12 - α 3 ) + λ 1 2 B 11 , κ 1 B ′ 22 = κ 1 2 B ′ 22 + κ 1 λ 1 ( 2 B ′ 12 - α 3 ) + λ 1 2 B 11 , κ 1 B ′ 23 = κ 1 3 B ′ 23 + κ 1 2 λ 1 ( B ′ 13 + 2 B ′ 22 - α 2 ) + κ 1 λ 1 2 ( 3 B ′ 12 - 2 α 1 ) + λ 1 3 B 11 , κ 1 B ′ 33 = κ 1 4 B ′ 33 + κ 1 3 λ 1 ( 4 B ′ 23 - α 1 ) + κ 1 2 λ 1 2 [ 2 ( 2 B ′ 22 + B ′ 13 ) - 3 α 2 ] + κ 1 λ 1 5 ( 4 B ′ 12 - 3 α 1 ) + λ 1 4 B 11 .$
$κ 1 B 111 = B 111 + 6 κ 1 λ 1 B 11 2 , κ 1 B 112 = κ 1 B 112 + λ 1 B 111 + 6 κ 1 λ 1 B 11 B 12 , κ 1 B 211 = κ 1 B 211 + λ 1 B 111 + 6 κ 1 λ 1 B 11 B 12 , κ 1 B 113 = κ 1 2 B 113 + 2 κ 1 λ 1 B 112 + λ 1 2 B 111 + κ 1 λ 1 ( 4 B 11 B 13 + 2 B 12 2 ) , κ 1 B 122 = κ 1 2 B 122 + 2 κ 1 λ 1 B 112 + λ 1 2 B 111 + κ 1 λ 1 ( 2 B 11 B 22 + 4 B 12 2 ) , κ 1 B 212 = κ 1 2 B 212 + κ 1 λ 1 ( B 211 + B 112 ) + λ 1 2 B 111 + κ 1 λ 1 ( 2 B 11 B 22 + 4 B 12 2 ) , κ 1 B 123 = κ 1 3 B 123 + κ 1 2 λ 1 ( B 113 + 2 B 112 ) + 3 κ 1 λ 1 2 B 112 + λ 1 3 B 111 + κ 1 λ 1 [ B 11 B 23 + B 12 ( 3 B 13 + 2 B 22 ) ] , κ 1 B 213 = κ 1 3 B 213 + κ 1 2 λ 1 ( B 113 + 2 B 212 ) + κ 1 λ 1 2 ( B 211 + 2 B 112 ) + λ 1 3 B 111 + κ 1 λ 1 [ B 11 B 23 + B 12 ( 3 B 13 + 2 B 22 ) ] , κ 1 B 222 = κ 1 3 B 222 + κ 1 2 λ 1 ( B 122 + 2 B 212 ) + κ 1 λ 1 2 ( B 211 + 2 B 112 ) + λ 1 3 B 111 + 6 κ 1 λ 1 B 12 B 22 , κ 1 B 133 = κ 1 4 B 133 + 4 κ 1 3 λ 1 B 123 + 2 κ 1 2 λ 1 2 ( B 113 + 2 B 122 ) + 4 κ 1 λ 1 3 B 112 + λ 1 4 B 111 + κ 1 λ 1 [ 2 B 13 2 + 4 B 12 B 23 ] , κ 1 B 223 = κ 1 4 B 223 + κ 1 3 λ 1 ( 2 B 222 + B 213 + B 123 ) + κ 1 2 λ 1 2 ( 3 B 212 + 2 B 122 + B 113 ) + κ 1 λ 1 3 ( B 211 + 3 B 112 ) + λ 1 4 B 111 + 2 κ 1 λ 1 [ B 12 B 23 + B 22 ( B 13 + B 22 ) ] , κ 1 B 233 = κ 1 5 B 233 + κ 1 4 λ 1 ( 4 B 223 + B 133 ) + κ 1 3 λ 1 2 ( 4 B 222 + 2 B 213 + 4 B 123 ) + κ 1 2 λ 1 3 ( 4 B 112 + 4 B 122 + 2 B 113 ) + κ 1 λ 1 4 ( B 211 + 4 B 112 ) + λ 1 5 B 111 + 2 κ 1 λ 1 B 23 ( B 13 + 2 B 22 )$
$B ′ 211 = λ 1 B 111 + B ′ 211 + 6 λ 1 B ′ 12 B 11 , B ′ 212 = λ 1 B 112 + λ 1 B ′ 211 + κ 1 B ′ 212 + 2 λ 1 B 11 ( κ 1 B ′ 22 + λ 1 B ′ 12 ) + 4 λ 1 B 12 B ′ 12 , B ′ 311 = λ 1 B 211 + λ 1 B ′ 211 + κ 1 B ′ 311 + 4 λ 1 B 11 ( κ 1 B ′ 13 + λ 1 B ′ 12 ) + 2 λ 1 B 12 B ′ 12 , B ′ 213 = λ 1 B 113 + λ 1 2 B ′ 211 + 2 κ 1 λ 1 B ′ 212 + κ 1 2 B ′ 213 + λ 1 B 11 ( κ 1 2 B ′ 23 + 2 κ 1 λ 1 B ′ 22 + λ 1 2 B ′ 12 ) + 2 λ 1 B 12 ( κ 1 B ′ 22 + λ 1 B ′ 12 ) + 3 λ 1 B 13 B ′ 12 , B ′ 222 = λ 1 B 122 + λ 1 2 B ′ 211 + 2 κ 1 λ 1 B ′ 212 + κ 1 2 B ′ 222 + 4 λ 1 B 12 ( κ 1 B ′ 22 + λ 1 B ′ 12 ) + 2 λ 1 B 22 B ′ 12 , B ′ 312 = λ 1 B 212 + λ 1 2 B ′ 211 + κ 1 λ 1 ( B ′ 212 + B ′ 311 ) + κ 1 2 B ′ 312 + λ 1 B 11 [ κ 1 2 B ′ 23 + κ 1 λ 1 ( B ′ 13 + B ′ 22 ) + λ 1 2 B ′ 12 ] + λ 1 B 12 [ κ 1 ( 3 B ′ 13 + B ′ 22 ) + 4 λ 1 B ′ 12 ] + λ 1 B 22 B ′ 12 , B ′ 223 = λ 1 B 123 + λ 1 3 B ′ 211 + 3 κ 1 λ 1 2 B ′ 212 + κ 1 2 λ 1 ( B ′ 213 + 2 B ′ 222 ) + κ 1 3 B ′ 223 + λ 1 B 12 ( κ 1 2 B ′ 23 + 2 κ 1 λ 1 B ′ 22 + λ 1 2 B ′ 12 ) + 2 λ 1 ( B 22 + B 13 ) ( κ 1 B ′ 22 + λ 1 B ′ 12 ) + λ 1 B 23 B ′ 12 , B ′ 313 = λ 1 B 213 + λ 1 3 B ′ 211 + κ 1 λ 1 2 ( 2 B ′ 212 + B ′ 311 ) + κ 1 2 λ 1 ( B ′ 213 + 2 B ′ 312 ) + κ 1 3 B ′ 313 + λ 1 B 12 [ 3 κ 1 2 B ′ 23 + 2 κ 1 λ 1 ( B ′ 13 + 2 B ′ 22 ) + 3 λ 1 2 B ′ 12 ] + 2 λ 1 B 13 ( κ 1 B ′ 13 + λ 1 B ′ 12 ) + λ 1 B 23 B ′ 12 , B ′ 322 = λ 1 B 222 + λ 1 3 B ′ 211 + κ 1 λ 1 2 ( 2 B ′ 212 + B ′ 311 ) + κ 1 2 λ 1 ( B ′ 222 + 2 B ′ 312 ) + κ 1 3 B ′ 322 + 2 λ 1 B 12 [ κ 1 2 B ′ 23 + κ λ ( B ′ 13 + B ′ 22 ) + λ 1 2 B ′ 12 ] + 2 λ 1 B 22 [ κ 1 ( B ′ 13 + B ′ 22 ) + 2 λ 1 B ′ 12 ] , B ′ 233 = λ 1 B 133 + λ 1 4 B ′ 211 + 4 κ 1 λ 1 3 B ′ 212 + 2 κ 1 2 λ 1 2 ( B ′ 213 + 2 B ′ 222 ) + 4 κ 1 3 λ 1 B ′ 223 + κ 1 4 B ′ 233 + 2 λ 1 B 13 ( κ 1 2 B ′ 23 + 2 κ 1 λ 1 B ′ 22 + λ 1 2 B ′ 12 ) + 4 λ 1 B 23 ( κ 1 B ′ 22 + λ 1 B ′ 12 ) , B ′ 323 = λ 1 B 223 + λ 1 4 B ′ 211 + κ 1 λ 1 3 ( 3 B ′ 212 + B ′ 311 ) + κ 1 2 λ 1 2 ( B ′ 213 + 2 B ′ 222 + 3 B ′ 312 ) + κ 1 3 λ 1 ( B ′ 223 + B ′ 313 + 2 B ′ 322 ) + κ 1 4 B ′ 323 + λ 1 B 22 [ 3 κ 1 2 B ′ 23 + 2 κ 1 λ 1 ( B ′ 13 + 2 B ′ 22 ) + 3 λ 1 2 B ′ 12 ] + λ 1 B 23 [ κ 1 ( B ′ 13 + B ′ 22 ) + 2 λ 1 B ′ 12 ] + λ 1 B 13 [ κ 1 2 B ′ 23 + κ 1 λ 1 ( B ′ 13 + B ′ 22 ) + λ 1 2 B ′ 12 ] , B ′ 333 = λ 1 B 233 + λ 1 5 B ′ 211 + λ 1 4 κ 1 ( 4 B ′ 212 + B ′ 311 ) + 2 κ 1 2 λ 1 3 ( 2 B ′ 232 + B ′ 213 + 2 B ′ 312 ) + 2 κ 1 3 λ 1 2 ( 2 B ′ 223 + B ′ 313 + 2 B ′ 322 ) + κ 1 4 λ 1 ( B ′ 233 + 4 B ′ 322 ) + κ 1 5 B ′ 333 + 2 λ 1 B 23 [ 3 κ 1 2 B ′ 23 + 2 κ 1 λ 1 ( B ′ 13 + 2 B ′ 22 ) + 3 λ 1 2 B ′ 12 ] .$
$κ 1 B ′ 211 = κ 1 B ′ 211 + λ 1 B 111 + 6 κ 1 λ 1 B 11 B ′ 12 , κ 1 B ′ 212 = κ 1 2 B ′ 212 + κ 1 λ 1 ( B ′ 211 + B 112 ) + λ 1 2 B 111 + 2 κ 1 λ 1 [ B 11 B ′ 22 + 2 B 12 B ′ 12 ] , κ 1 B ′ 311 = κ 1 2 B ′ 311 + κ 1 λ 1 ( B ′ 211 + B 211 ) + λ 1 2 B 111 + 2 κ 1 λ 1 [ 2 B 11 B ′ 13 + B 12 B ′ 12 ] , κ 1 B ′ 213 = κ 1 3 B ′ 213 + κ 1 2 λ 1 ( B 113 + 2 B ′ 212 ) + κ 1 λ 1 2 ( 2 B 112 + B ′ 211 ) + λ 1 3 B 111 + κ 1 λ 1 [ B 11 B ′ 23 + 2 B 12 B ′ 22 + 3 B 13 B ′ 12 ] , κ 1 B ′ 222 = κ 1 3 B ′ 222 + κ 1 2 λ 1 ( B 122 + 2 B ′ 212 ) + κ 1 λ 1 2 ( 2 B 112 + B ′ 211 ) + λ 1 3 B 111 + 2 κ 1 λ 1 [ 2 B 12 B ′ 22 + B 22 B ′ 12 ] , κ 1 B ′ 312 = κ 1 3 B ′ 312 + κ 1 2 λ 1 ( B 212 + B ′ 212 + B ′ 311 ) + κ 1 λ 1 2 ( B 211 + B 112 + B ′ 211 ) + λ 1 3 B 111 + κ 1 λ 1 [ B 11 B ′ 23 + B 12 ( B ′ 22 + 3 B ′ 13 ) + B 22 B ′ 12 ] , κ 1 B ′ 223 = κ 1 4 B ′ 223 + κ 1 3 λ 1 ( B 123 + 2 B ′ 222 + B ′ 213 ) + κ 1 2 λ 1 2 ( 2 B 122 + B 113 + 3 B ′ 212 ) + κ 1 λ 1 3 ( 3 B 112 + B ′ 211 ) + λ 1 4 B 111 + κ 1 λ 1 [ B 12 B ′ 23 + 2 ( B 13 + B 22 ) B ′ 22 + B 23 B ′ 12 ] , κ 1 B ′ 313 = κ 1 4 B ′ 313 + κ 1 3 λ 1 ( B 213 + B ′ 213 + 2 B ′ 312 ) + κ 1 2 λ 1 2 ( 2 B 212 + 2 B ′ 212 + B 113 + B ′ 311 ) + κ 1 λ 1 3 ( 2 B 112 + B 211 + B ′ 211 ) + λ 1 4 B 111 + κ 1 λ 1 [ 3 B 12 B ′ 23 + 2 B 13 B ′ 13 + B 23 B ′ 12 ] , κ 1 B ′ 322 = κ 1 4 B ′ 322 + κ 1 3 λ 1 ( B 222 + B ′ 222 + 2 B ′ 312 ) + κ 1 2 λ 1 2 ( B 122 + 2 B 212 + 2 B ′ 212 + B ′ 311 ) + κ 1 λ 1 3 ( 2 B 112 + B 211 + B ′ 211 ) + λ 1 4 B 111 + 2 κ 1 λ 1 [ B 12 B ′ 23 + B 22 ( B ′ 13 + B ′ 22 ) ] , κ 1 B ′ 233 = κ 1 5 B ′ 233 + κ 1 4 λ 1 ( B 133 + 4 B ′ 223 ) + κ 1 3 λ 1 2 ( 4 B 123 + 2 B ′ 213 + 4 B ′ 222 ) + κ 1 2 λ 1 3 ( 4 B 122 + 2 B 113 + 4 B ′ 212 ) + κ 1 λ 1 4 ( 4 B 112 + B ′ 211 ) + λ 1 5 B 111 + 2 κ 1 λ 1 [ B 13 B ′ 23 + 2 B 23 B ′ 22 ] , κ 1 B ′ 323 = κ 1 5 B ′ 323 + κ 1 4 λ 1 ( B 223 + B ′ 223 + B ′ 313 + 2 B ′ 322 ) + κ 1 3 λ 1 2 ( B 123 + 2 B 222 + B 213 + B ′ 213 + 2 B ′ 222 + 3 B ′ 312 ) + κ 1 2 λ 1 3 ( B 113 + 2 B 122 + 3 B 212 + 3 B ′ 212 + B ′ 311 ) + κ 1 λ 1 4 ( 3 B 112 + B 211 + B ′ 211 ) + λ 1 5 B 111 + κ 1 λ 1 [ B 13 B ′ 23 + 3 B 22 B ′ 23 + B 23 ( B ′ 13 + B ′ 22 ) ] , κ 1 B ′ 333 = κ 1 6 B ′ 333 + κ 1 5 λ 1 ( B 233 + B ′ 233 + 4 B ′ 323 ) + κ 1 4 λ 1 2 ( B 133 + 4 B 223 + 4 B ′ 223 + 2 B ′ 313 + 4 B ′ 322 ) + 2 κ 1 3 λ 1 3 ( 2 B 123 + B 213 + 2 B 222 + B ′ 213 + 2 B ′ 222 + 2 B ′ 312 ) + κ 1 2 λ 1 4 ( 4 B 122 + 2 B 113 + 4 B 212 + 4 B ′ 212 + B ′ 311 ) + κ 1 λ 1 5 ( 4 B 112 + B 211 + B ′ 211 ) + λ 1 6 B 111 + 6 κ 1 λ 1 B 23 B ′ 23 .$
$κ 1 C ′ 112 = κ 1 C ′ 112 + λ 1 C 111 + 6 κ 1 λ 1 B 11 B 12 , κ 1 C ′ 113 = κ 1 2 C ′ 113 + κ 1 λ 1 ( 2 C ′ 112 - γ 5 ) + λ 1 2 C 111 + 2 κ 1 λ 1 ( B 12 2 + 2 B 11 B 13 ) , κ 1 C ′ 122 = κ 1 2 C ′ 122 + κ 1 λ 1 ( 2 C ′ 112 - γ 5 ) + λ 1 2 C 111 + 2 κ 1 λ 1 ( 2 B 12 2 + B 11 B 22 ) , κ 1 C ′ 123 = κ 1 3 C ′ 123 + κ 1 2 λ 1 ( C ′ 113 + 2 C ′ 122 - γ 4 ) + κ 1 λ 1 2 ( 3 C ′ 112 - 2 γ 5 ) + λ 1 3 C 111 + κ 1 λ 1 [ B 11 B 23 + B 12 ( 3 B 13 + 2 B 22 ) ] , κ 1 C ′ 222 = κ 1 3 C ′ 222 + κ 1 2 λ 1 ( 3 C ′ 122 - γ 4 ) + κ 1 λ 1 2 ( 3 C ′ 112 - 2 γ 5 ) + λ 1 3 C 111 + 6 κ 1 λ 1 B 12 B 22 , κ 1 C ′ 133 = κ 1 4 C ′ 133 + κ 1 3 λ 1 ( 4 C ′ 123 - γ 3 ) + κ 1 2 λ 1 2 ( 2 C ′ 113 + 4 C ′ 122 - 3 γ 4 ) + κ 1 λ 1 3 ( 4 C ′ 112 - 3 γ 5 ) + λ 1 4 C 111 + 2 κ 1 λ 1 ( B 13 2 + 2 B 12 B 23 ) , κ 1 C ′ 223 = κ 1 4 C ′ 223 + κ 1 3 λ 1 ( 2 C ′ 123 + 2 C ′ 222 - γ 3 ) + κ 1 2 λ 1 2 ( C ′ 113 + 5 C ′ 122 - 3 γ 4 ) + κ 1 λ 1 3 ( 4 C ′ 112 - 3 γ 5 ) + λ 1 4 C 111 + 2 κ 1 λ 1 [ B 12 B 23 + B 22 ( B 13 + B 22 ) ] , κ 1 C ′ 233 = κ 1 5 C ′ 233 + κ 1 4 λ 1 ( 4 C ′ 223 + C ′ 133 - γ 2 ) + κ 1 3 λ 1 2 ( 8 C 123 + 4 C ′ 222 - 6 γ 3 ) + κ 1 2 λ 1 3 ( 2 C ′ 113 + 8 C ′ 122 - 6 γ 4 ) + κ 1 λ 1 4 ( 5 C ′ 112 - 4 γ 5 ) + λ 1 5 C 111 + 2 κ 1 λ 1 B 23 [ B 13 + 2 B 22 ] , κ 1 C ′ 333 = κ 1 6 C ′ 333 + κ 1 5 λ 1 ( 6 C ′ 233 - γ 1 ) + κ 1 4 λ 1 2 ( 12 C ′ 223 + 3 C ′ 133 - 5 γ 2 ) + κ 1 3 λ 1 3 ( 12 C ′ 123 + 8 C ′ 222 - 10 γ 3 ) + κ 1 2 λ 1 4 ( 12 C ′ 122 + 3 C ′ 113 - 10 γ 4 ) + κ 1 λ 1 5 ( 6 C ′ 112 - 5 γ 5 ) + λ 1 6 C 111 + 6 κ 1 λ 1 B 23 2 .$