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
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
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
1
þ
˃
d
e 2
2 þ
e 4
8
cos 2 u
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
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
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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