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A comparison of the coecients of the two polynomials Δx ( Δα )and Δy ( Δα )
constitutes the fifth step:
Δx (meta-equator)
Δα : q 1 b 1 α 1 + l 1 β 1 = s 1 .
(16.67)
Δα 2 :( q 1 b 2 + q 2 b 1 ) α 1 + l 2 β 1 +( q 1 b 1
l 1 ) α 2 +2 q 1 b 1 l 1 β 2 = s 2 .
Δy (meta-equator)
Δα : −l 1 α 1 + q 1 b 1 β 1 =0 ,
Δα 2 :
l 2 α 1 +( q 1 b 2 + q 2 b 1 ) β 1
2 q 1 b 1 l 1 α 2 +( q 1 b 1
l 1 ) β 2 =0 .
(16.68)
A matrix version of the above equations is
q 1 b 1
l 1
α 1
β 1
α 2
β 2
s 1
s 2
0
0
0
0
q 1 b 2 + q 2 b 1
l 2
q 1 b 1 − l 1 2 q 1 b 1 l 1
=
.
(16.69)
−l 1
q 1 b 1
0
0
l 2
q 1 b 2 + q 1 b 1
2 q 1 b 1 l 1 q 1 b 1
l 1
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 1 b 1 l 1 + l 1 ) β 1 = s 1 1 = q 1 b 1 l 1 β 1
(16.70)
β 1 = l 1 s 1 ( q 1 b 1 + l 1 ) 1 ,
α 1 = q 1 b 1 s 1 ( q 1 b 1 + l 1 ) 1 .
2nd row, 3rd row
( q 1 b 2 + q 2 b 1 ) α 1 + l 2 β 1 +( q 1 b 1
l 1 ) α 2
2 q 1 b 1 l 1 β 2 = s 2 ,
l 2 α 1 +( q 1 b 2 + q 2 b 1 ) β 1
2 q 1 b 1 l 1 α 2 +( q 1 b 1
l 1 ) β 2 =0
( q 1 b 2 + q 2 b 1 ) q 1 b 1 s 1 ( q 1 b 1 + l 1 ) 1 + l 2 l 1 s 1 ( q 1 b 1 + l 1 ) 1
s 1 =
(16.71)
=( l 1
q 1 b 1 ) α 2
2 q 1 b 1 l 1 β 2 ,
( q 1 b 2 + q 2 b 1 ) l 1 s 1 ( q 1 b 1 + l 1 ) 1
− l 2 q 1 b 1 s 1 ( q 1 b 1 + l 1 ) 1 =
=2 q 1 b 1 l 1 α 2 +( l 1
q 1 b 1 ) β 2 ,
q 1 b 1
α 2
β 2
=
l 1 2 q 1 b 1 l 1
2 q 1 b 1 l 1 q 1 b 1
l 1
= s 2
( q 1 b 1 + l 1 ) 1 l 1 l 2 s 1
( q 1 b 2 + q 2 b 1 )( q 1 b 1 + l 1 ) 1 q 1 b 1 s 1
( q 1 b 1 + l 1 ) 1 l 2 q 1 b 1 s 1
( q 1 b 1 + l 1 ) 1 ( q 1 b 2 + q 2 b 1 ) l 1 s 1
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