Geoscience Reference
In-Depth Information
Solution of Bessel's Differential Equation
The Relationship Between S and
˃
Integrating (
5.85
) gives the relationship between S and
˃
. In order to find the
integral, we shall first convert u into a function of
as shown in Fig.
5.43
.We
extend the arc of the great circle P
1
0
P
2
0
to intersect the equator of the auxiliary
sphere at point P
0
0
. The azimuth of P
1
0
P
2
0
at the point P
0
0
is m, and the arc of the
great circle P
0
0
P
1
0
ﾼ
˃
M. Obviously, when point P
1
0
and the arc of the great circle
P
1
0
P
2
0
are given, the values of m and M are also defined. So, the purpose of
extending the arc of the great circle P
1
0
P
2
0
is to find the spherical triangle that is
relevant to the spherical quadrangle in order to apply the formula for spherical
triangles to find the solution.
We assume that P
0
is a moving point along the arc P
1
0
P
2
0
. When P
0
moves, the
between P
0
and P
1
0
and the spherical latitude u of point P
0
also change
accordingly; hence, the relationship between u and
distance
˃
can be established.
Considering P
2
0
as the moving point P
0
, from the right-angled spherical triangle
P
0
0
Q
2
P
0
2
, one obtains:
˃
sin u
ﾼ
cos m sin M
ð
þ ˃
Þ
or
cos
2
u
cos
2
m sin
2
M
ﾼ
1
ð
þ ˃
Þ:
ð
5
:
87
Þ
Substituting the above equation into (
5.85
) gives:
p
1
dS
ﾼ
a
e
2
þ
e
2
cos
2
m sin
2
M
ð
þ ˃
Þ
d
˃
s
1
a
p
e
2
e
2
cos
2
m sin
2
M
ﾼ
1
e
2
þ
ð
þ ˃
Þ
d
˃
1
q
1
e
0
2
cos
2
m sin
2
M
ﾼ
b
þ
ð
þ ˃
Þ
d
˃
,
which can be written as:
q
1
k
2
sin
2
M
dS
ﾼ
b
þ
ð
þ ˃
Þ
d
˃
,
ð
5
:
88
Þ
where k
2
e
02
cos
2
m.
In order to find the integral of (
5.88
), we expand the integrand into a series of
ﾼ
˃
as follows:
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