16-3 The equations of the oblique Mercator projection 375
1st row, 3rd row
⇒
q 1 b 1 α 1 + l 1 β 1 = s 1 , −l 1 α 1 + q 1 b 1 β 1 = 0
⇒
(q
2
1 b
2
1 l
−1
1 + l 1 )β 1 = s 1 , α 1 = q 1 b 1 l
−1
1 β 1
⇒
β 1 = l 1 s 1 (q
2
1 b
2
1 + l
2
1 )
−1 ,
α 1 = q 1 b 1 s 1 (q
2
1 b
2
1 + l
2
1 )
−1 .
(16.70)
2nd row, 3rd row
⇒
(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 ,
−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
⇒
(q 1 b 2 + q 2 b
2
1 )q 1 b 1 s 1 (q
2
1 b
2
1 + l
2
1 )
−1 + l 2 l 1 s 1 (q
2
1 b
2
1 + l
2
1 )
−1
− s 1 =
= (l
2
1 − q
2
1 b
2
1 )α 2 − 2q 1 b 1 l 1 β 2 ,
(q 1 b 2 + q 2 b
2
1 )l 1 s 1 (q
2
1 b
2
1 + l
2
1 )
−1
− l 2 q 1 b 1 s 1 (q
2
1 b
2
1 + l
2
1 )
−1 =
= 2q 1 b 1 l 1 α 2 + (l
2
1 − q
2
1 b
2
1 )β 2 ,
(16.71)
q
2
1 b
2
1 − l
2
1
2q 1 b 1 l 1
−2q 1 b 1 l 1 q
2
1 b
2
1 − l
2
1
α 2
β 2
=
=
s 2 − (q
2
1 b
2
1 + l
2
1 )
−1 l 1 l 2 s 1 − (q 1 b 2 + q 2 b
2
1 )(q
2
1 b
2
1 + l
2
1 )
−1 q 1 b 1 s 1
(q
2
1 b
2
1 + l
2
1 )
−1 l 2 q 1 b 1 s 1 − (q
2
1 b
2
1 + l
2
1 )
−1 (q 1 b 2 + q 2 b
2
1 )l 1 s 1
⇒
α 2 =
1
(q 2
1 b 2
1 + l 2
1 ) 3 ×
×
s 2 (q
2
1 b
2
1 − l
2
1 )(q
2
1 b
2
1 + l
2
1 ) + s 1 [(q 1 b 2 + q 2 b
2
1 )(3l
2
1 − q
2
1 b
2
1 )q 1 b 1 − l 1 l 2 (3q
2
1 b
2
1 − l
2
1 )]
,
β 2 =
1
(q 2
1 b 2
1 + l 2
1 ) 3 ×
×
2q 1 b 1 l 1 (q
2
1 b
2
1 + l
2
1 )s 2 + s 1 [(q 1 b 2 + q 2 b
2
1 )(−3q
2
1 b
2
1 + l
2
1 )l 1 + (−3l
2
1 + q
2
1 b
2
1 )q 1 b 1 l 1 ]
.
(16.72)
End of Proof.