Geoscience Reference
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meridian is quite small, generally within 0-3.5
. Besides, the arc value
l
ρ
is a small
quantity, so we can expand (
6.19
) at point (0, q) according to a Taylor series
expansion of binary functions. In this case, the increments of the independent
variable of an arbitrary point (l, q) relative to the expansion point (0, q) are l and
0, respectively, i.e., the increment of q relative to the expansion point is 0. Hence,
the partial derivative with respect to q and the term that includes mixed partial
derivatives with respect to q are all 0. The x series is expanded as:
0;ðÞ
0;ðÞ
0;ðÞ
0;ðÞ
2
f
1
∂
3
f
1
∂
4
f
1
∂
ðÞþ
∂
f
1
∂
1
2
∂
1
3
∂
1
4
∂
l
2
l
3
l
4
x
ﾼ
f
1
0
;
q
l
þ
þ
þ
þ ...
l
!
l
2
!
l
3
!
l
4
m
2
l
2
m
3
l
3
m
4
l
4
ﾼ
m
0
þ
m
1
l
þ
þ
þ
þ ...
,
which is the power series of the difference in longitude l. The value of each
partial derivative at point (0, q) no longer includes the variable l. Hence m
i
(i
ﾼ
0,
1, 2, 3, 4,....) is the function of isometric latitude q; m
0
ﾼ
f
1
(0, q) indicates the
x coordinate of the point (0, q) on the central meridian. From the third condition for
the Gauss projection (when l
ﾼ
0, x
ﾼ
X), one can get m
0
ﾼ
X.
By the same token, we have:
0;ðÞ
0;ðÞ
0;ðÞ
0;ðÞ
2
f
2
∂
3
f
2
∂
4
f
2
∂
ðÞþ
∂
f
2
∂
1
2
∂
1
3
∂
1
4
∂
l
2
l
3
l
4
y
ﾼ
f
2
0
;
q
l
þ
þ
þ
þ ...
l
2
l
3
l
4
l
!
!
!
n
2
l
2
n
3
l
3
n
4
l
4
ﾼ
n
0
þ
n
1
l
þ
þ
þ
þ ...
,
which is the power series of the difference in longitude l. n
i
(i
ﾼ
0, 1, 2, 3, 4,....) is
the function of isometric latitude q; n
0
ﾼ
f
2
(0, q) indicates the y coordinate of the
point (0, q) on the central meridian. According to the second condition for Gauss
projection (when l
0.
Equation (
6.19
) has been expanded into the power series of the longitude
difference l:
ﾼ
0, y
ﾼ
0), we have n
0
ﾼ
,
m
2
l
2
m
3
l
3
m
4
l
4
x
ﾼ
X
þ
m
1
l
þ
þ
þ
þ ...
ð
6
:
20
Þ
n
2
l
2
n
3
l
3
n
4
l
4
y
ﾼ
n
1
l
þ
þ
þ
þ ...
where m
1
, m
2,
...
are undetermined coefficients. They are the functions
of the isometric latitude q (or geodetic latitude B). To apply the first condition for
Gauss projection, taking the partial derivative of (
6.20
) yields:
, n
1
, n
2
,
...
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