372 16 “Ellipsoid-of-revolution to cylinder”: oblique aspect
The relations (16.44) together with the relations (16.50) lead to the relations (16.51). Let us prove
the other central relations here.
Proof: (16.47): b 1 .
tan B =
A
2
1 − E 2 sin i
sin α
A
1
2 + (A
2
2 cos 2 i − A
1
2 ) sin
2 α
,
d tan B
dα
=
1
cos 2 B
dB
dα
=
A
2
1 − E 2 sin i
A
1
2 cos α
[A
1
2 + (A
2
2 cos 2 i − A
1
2 ) sin
2 α] 3/2
,
dB
dα
= cos
2 B
d tan B
dα
=
1
1 + tan
2 B
d tan B
dα
=
= [A
2 A
1
2 (1 − E
2 ) sin i cos α][A
1
2 + (A
2
2 cos
2 i − A
1
2 ) sin
2 α]
−1/2
×
×[A
1
2 (1 − E
2 )
2 + E
2 A
2
2 sin
2 i sin
2 α]
−1
⇒
b 1 :=
dB
dα
(α 0 ) .
(16.59)
End of Proof.
Proof: (16.48): l 1 .
tan(L − Ω) =
A
2
A
1
cos i tan α ,
(16.60)
d tan(L − Ω)
dα
=
1
cos 2 (L − Ω)
dL
dα
=
A
2
A
1
cos i
cos 2 α
,
cos
2 (L − Ω) =
1
1 + tan
2 (L − Ω)
=
1
1 +
A
2
2 cos 2 i
A
1
2
tan
2 α
⇒
dL
dα
=
A
2
A
1
cos i
cos 2 α
=
1
1 +
A
2
2 cos 2 i
A
1
2
tan
2 α
=
=
A
1 A
2 cos i
A
1
2 cos 2 α + A
2
2 cos 2 i sin
2 α
⇒
l 1 :=
dL
dα
(α 0 ) .
(16.61)
End of Proof.