Geoscience Reference
In-Depth Information
The Relationship Between l and
ʻ
Integrating (
5.86
) gives the relationship between l and
ʻ
. As stated above, u has
been converted into the function of
˃
,soifd
˃
is used to represent d
ʻ
, the differential
equation (
5.86
) can be solved.
It follows from (
5.82
) that:
sin A
cos u
d
d
ʻ ᄐ
˃:
Also, let P
2
0
be a moving point P
0
; from the right triangle P
0
0
Q
2
P
2
0
we have:
sin m
cos u
,
sin A
ᄐ
and we obtain:
sin m
cos
2
u
d
d
ʻ ᄐ
˃:
Substituting the above equation into (
5.86
) yields:
p
1
sin m
cos
2
u
d
dl
ᄐ
e
2
cos
2
u
˃:
In order to find the integral, we expand the above equation into the series of u,
hence:
sin m
cos
2
u
d
e
2
2
e
4
8
cos
2
u
cos
4
u
dl
ᄐ
1
þ
˃
d
e
2
2
þ
e
4
8
cos
2
u
ᄐ
d
ʻ
sin m
þ
˃:
Substituting (
5.87
) into the above equation gives:
2
3
sin m
e
2
e
4
8
e
4
8
4
5
d
cos
2
m sin
2
M
dl
ᄐ
d
ʻ
2
þ
ð
þ ˃
Þþ
˃
2
3
sin m
e
2
e
4
8
e
4
16
cos
2
m
e
4
16
cos
2
m cos 2 M
4
5
d
ᄐ
d
ʻ
2
þ
þ
ð
þ ˃
Þþ
˃:
Integrating the above equation (accurate to the term of e
4
) results in:
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