374 16 “Ellipsoid-of-revolution to cylinder”: oblique aspect
In a third step, the boundary condition in the above form is represented in the
meta-longitude dependence by means of (16.47)–(16.49)
(where in a fourth step we identify (16.45) by (16.46)):
∆x(meta-equator) =
= α 1 (q 1 b 1 ∆α + q 1 b 2 ∆α
2 + q 2 b
2
1 ∆α
2 ) + β 1 (l 1 ∆α + l 2 ∆α
2 )+
+α 2 (q
2
1 b
2
1 ∆α
2
− l
2
1 ∆α
2 ) + β 2 2q 1 b 1 l 1 ∆α
2 + O x3 =
= s 1 ∆α + s 2 ∆α
2 ,
∆y(meta-equator) =
−α 1 (l 1 ∆α + l 2 ∆α
2 ) + β 1 (q 1 b 1 ∆α + q 1 b 2 ∆α
2 + q 2 b
2
1 ∆α
2 )−
−α 2 2q 1 b 1 l 1 ∆α
2 + β 2 (q
2
1 b
2
1 ∆α
2
− l
2
1 ∆α
2 ) + O y3 = 0 .
(16.66)
A comparison of the coefficients of the two polynomials ∆x(∆α) and ∆y(∆α)
constitutes the fifth step:
∆x(meta-equator)
⇒
∆α : q 1 b 1 α 1 + l 1 β 1 = s 1 ,
∆α
2 : (q 1 b 2 + q 2 b
2
1 )α 1 + l 2 β 1 + (q
2
1 b
2
1 − l
2
1 )α 2 + 2q 1 b 1 l 1 β 2 = s 2 .
(16.67)
∆y(meta-equator)
⇒
∆α : −l 1 α 1 + q 1 b 1 β 1 = 0 ,
∆α
2 : −l 2 α 1 + (q 1 b 2 + q 2 b
2
1 )β 1 − 2q 1 b 1 l 1 α 2 + (q
2
1 b
2
1 − l
2
1 )β 2 = 0 .
(16.68)
A matrix version of the above equations is
⎡
⎢
⎢
⎢
⎢
⎢
⎣
q 1 b 1
l 1
0
0
q 1 b 2 + q 2 b
2
1
l 2
q
2
1 b
2
1 − l
2
1
2q 1 b 1 l 1
−l 1
q 1 b 1
0
0
−l 2
q 1 b 2 + q 2 b
2
1 −2q 1 b 1 l 1 q
2
1 b
2
1 − l
2
1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎣
α 1
β 1
α 2
β 2
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
s 1
s 2
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
(16.69)