16-3 The equations of the oblique Mercator projection 373
Proof: (16.47): b 2 .
d
2 B
dα 2 = A
2 A
1
2 (1 − E
2 ) sin i ×
×
− sin α[A
1
2 (1 − E
2 )
2 + E
2 A
2
2 sin
2 i sin
2 α]
−1 [A
1
2 + (A
2
2 cos
2 i − A
1
2 ) sin
2 α]
−1/2
−
−2E
2 A
2
2 sin
2 i sin α cos
2 α×
×[A
1
2 (1 − E
2 )
2 + E
2 A
2
2 sin
2 i sin
2 α]
−2 [A
1
2 + (A
2
2 cos
2 i − A
1
2 ) sin
2 α]
−1/2
−
−(A
2
2 cos
2 i − A
1
2 ) sin α cos
2 α×
×[A
1
2 (1 − E
2 )
2 + E
2 A
2
2 sin
2 i sin
2 α]
−1 [A
1
2 + (A
2
2 cos
2 i − A
1
2 ) sin
2 α]
−3/2
⇒
2b 2 :=
d
2 B
dα 2 (α 0 ) .
(16.62)
End of Proof.
Proof: (16.48): l 2 .
d
2 L
dα 2 = −2
A
1 A
2 cos i(A
2
2 cos
2 i − A
1
2 ) sin α cos α
(A
1
2 cos 2 α + A
2
2 cos 2 i sin
2 α) 2
,
2l 2 :=
d
2 L
dα 2 (α 0 ) .
(16.63)
End of Proof.
Proof: (16.52)–(16.54).
In a first step, (16.44) is specified by
∆x(meta-equator) =
= α 1 ∆q + β 1 ∆l + α 2 (∆q
2
− ∆l
2 ) + β 2 2∆q∆l + O x3 ,
∆y(meta-equator) =
= β 1 ∆q − α 1 ∆l + β 2 (∆q
2
− ∆l
2 ) − α 2 2∆q∆l + O y3 = 0 .
(16.64)
Implementation of (16.50) constitutes the second step:
∆x(meta-equator) =
= α 1 q 1 ∆b + α 1 q 2 ∆b
2 + β 1 ∆l + α 2 (q
2
1 ∆b
2
− ∆l
2 ) + β 2 2q 1 ∆b∆l + O x3 ,
∆y(meta-equator) =
= β 1 q 1 ∆b + β 1 q 2 ∆b
2
− α 1 ∆l + β 2 (q
2
1 ∆b
2
− ∆l
2 ) − α 2 2q 1 ∆b∆l + O y3 = 0 .
(16.65)