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l 1 ) ,
l 1 l 2 (3 q 1 b 1
1
( q 1 b 1 + l 1 ) 3 ×
β 2 =
(16.54)
× 2 q 1 b 1 l 1 ( q 1 b 1 + l 1 ) s 2 + s 1 ( q 1 b 2 + q 2 b 1 )( 3 q 1 b 1 + l 1 ) l 1
+(
3 l 1 + q 1 b 1 ) q 1 b 1 l 1 .
End of Theorem.
The coecients
{
q 1 ,q 2
}
,
{
s 1 ,s 2
}
,
{
b 1 ,b 2
}
,and
{
l 1 ,l 2
}
are collected in the following Boxes 16.2 -
16.5 .
Box 16.2 (Isometric latitude q ( b ) as a function of latitude b ).
Power series expansion Δq = N
r =1 q r Δb r up to order N =2
(higher-order terms are given by Engels and Grafarend 1995 ):
E 2
1
q 1 :=
E 2 sin 2 B 0 ) ,
(16.55)
cos B 0 (1
sin B 0
2cos 2 B 0 (1
E 2 sin 2 B 0 ) 2 [1 + E 2 (1
3sin 2 B 0 )+ E 4 (
2+3sin 2 B 0 )] .
q 2 :=
Box 16.3 (Arc length of the oblique meta-equator).
Power series expansion Δs = N
r =1 s r Δα r up to order N =2:
s 1 ( α 0 ):= A 1 1
E 2 cos 2 α 0 ,
A 1 E 2 sin α 0 cos α 0
s 2 ( α 0 ):= 1
2
1
E 2 cos 2 α 0 .
(16.56)
Box 16.4 (Latitude B ( α ) as a function of meta-longitude α ).
Power series expansion Δb = N
r =1 b r Δα r uptoorderN =2:
A 2 A 1 (1
E 2 )sin i cos α 0
b 1 :=
E 2 ) 2 + E 2 A 2 sin 2 i sin 2 α 0 ]
[ A 1 +( A 2 cos 2 i
[ A 1 (1
A 1 )sin 2 α 0 ] 1 / 2 ,
A 2 A 1 (1 − E 2 )sin i
2 b 2 :=
E 2 ) 2 + E 2 A 2 sin 2 i sin 2 α 0 ] 2
[ A 1 +( A 2 cos 2 i
[ A 1 (1
A 1 )sin 2 α 0 ] 3 / 2
×
(16.57)
 
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