It follows (2.208, 2.209) that
x
2 + y
2 = R g
2 + ρ h
2 + 2 χ g h
� 2
� 2
x + y = R g
� 2 + ρ h
� 2 + 2 χ g
� h
�
2 (x y
� − x
� y) = 2 (g h
� − g
� h) σ = 2 p
−1 k σ. (2.210)
From (2.204, 2.210) we express the perturbation m 4 in terms of
the new variable set as
m 4 =
L (R g
2 + ρ h
2 + 2 χ g h)
2
+ M (R g
2 + ρ h
2 + 2 χ g h) (R g
� 2 + ρ h
� 2 + 2 χ g
� h
� )
+ N (R g
� 2 + ρ h
� 2 + 2 χ g
� h
� )
2
+ P (2 σ k p
−1 ) (R g
2 + ρ h
2 + 2 χ g h)
+ Q (2 σ k p
−1 ) (R g
� 2 + ρ h
� 2 + 2 χ g
� h
� )
+ K [ 4 σ
2 k
2 p
−2 (ρ R − χ
2 ) ],
(2.211)
where we have made use in the K-term of the readily verifiable
fact that (2.209)
σ
2 = ρ R − χ
2 .
(2.212)
Expanding (2.211) and collecting terms, we now form
z I
m 4 dz = A R
2 + B ρ
2 + C χ
2 + D R ρ + E R χ
z O
+F ρ χ + e R σ + f ρ σ + c χ σ,
(2.213)
where we have defined new coefficients A, . . . , c. Collecting terms
in (2.213) according to the definition (2.206), the new coefficients
are
A =
z I
(L g
4 + M g
2 g
� 2 + N g
� 4 ) dz
z O
B =
z I
(L h
4 + M h
2 h
� 2 + N h
� 4 ) dz
z O
C =
z I
(4 L g
2 h
2 + 4 M g h g
� h
� + 4 N g
� 2 h
� 2 − 4 K k
2 p
−2 ) dz
z O
D =
z I
[ 2 L g
2 h
2 + M (g
2 h
� 2 + h
2 g
� 2 ) + 2 N g
� 2 h
� 2
z O
+4 K k
2 p
−2 ] dz
72
Chapter 2. Geometrical optics