/ 5 5 \ / 21 \
M/b = ( 1 + n + -n squared + -n cubed) [alpha]0 - ( 3n + 3n squared + --n cubed) [alpha]1
\ 4 4 / \ 8 /
/15 15 \ /35 \
+ ( --n squared + --n cubed) [alpha]2 - ( --n cubed ) [alpha]3,
\ 8 8 / \24 /
where
[alpha]0 = [phi]2 - [phi]1
[alpha]1 = sin ([phi]2 - [phi]1) cos ([phi]2 + [phi]1)
[alpha]2 = sin 2([phi]2 - [phi]1) cos 2([phi]2 + [phi]1)
[alpha]3 = sin 3([phi]2 - [phi]1) cos 3([phi]2 + [phi]1).
The part of M which depends on n cubed is very small; in fact, if we
calculate it for one of the longest arcs measured, the Russian arc, it
amounts to only an inch and a half, therefore we omit this term, and put
for M/b the value
/ 5 \ /15 \
(l + n + --n squared) [alpha]0 - (3n + 3n squared) [alpha]1 + ( --n squared) [alpha]2.
\ 4 / \ 8 /
Now, if we suppose the observed latitudes to be affected with errors,
and that the true latitudes are [phi]1 + x1, [phi]2 + x2; and if further
we suppose that n1 + dn is the true value of a - b:a + b, and that n1
itself is merely a very approximate numerical value, we get, on making
these substitutions and neglecting the influence of the corrections x on
the _position_ of the arc in latitude, i.e. on [phi]1 + [phi]2,
/ 5 \ / \ /15 \
M/b = ( 1 + n + --n1 squared)[alpha]0 - (3n1 + 3n1 squared)[alpha]1 + ( --n1 squared)[alpha]2
\ 4 / \ / \8 /
_ _
| / 5 \ / \ /15 \ |
+ | ( 1 + -- n1 )a0 - ( 3 + 6n1 )a1 + ( -- n1 )a2 | dn
|_ \ 2 / \ / \4 / _|
_ _
| da1 |
+ | 1 + n1 - 3n --- | da0;
|_ da0_|
here da0 = x2 - x1; and as b is only known approximately, put b = b1(1 +
u); then we get, after dividing through by the coefficient of da0, which
is = 1 + n1 - 3n1 cos([phi]2 - [phi]1) cos([phi]2 + [phi]1), an equation
of the form x2 = x1 + h + fu + gv, where for convenience we put v for
dn.
Now in every measured arc there are not only the extreme stations
determined in latitude, but also a number of intermediate stations so
that if there be i + 1 stations there will be i equations
x2 = x1 + f1u + g1v + h1
x3 = x1 + f2u + g2v + h2
: : :
: : :
x_i = x1 + f_iu + g_iv + h_i
In combining a number of different arcs of meridian, with the view of
determining the figure of the earth, each arc will supply a number of
equations in u and v and the corrections to its observed latitudes.
Then, according to the method of least squares, those values of u and v
are the most probable which render the sum of the squares of _all_ the
errors x a minimum. The corrections x which are here applied arise not
from errors of observation only. The mere uncertainty of a latitude, as
determined with modern instruments, does not exceed a very small
fraction of a second as far as errors of observation go, but no accuracy
in observing will remove the error that may arise from local attraction.
This, as we have seen, may amount to some seconds, so that the
corrections x to the observed latitudes are attributable to local
attraction. Archdeacon Pratt objected to this mode of applying least
squares first used by Bessel; but Bessel was right, and the objection is
groundless. Bessel found, in 1841, from ten meridian arcs with a total
amplitude of 50 deg..6:
a = 3272077 toises = 6377397 metres.
e (ellipticity) = (a - b)/a = 1/299.15 (prob. error +- 3.2).
The probable error in the length of the earth's quadrant is +- 336 m.
We now give a series of some meridian-arcs measurements, which were
utilized in 1866 by A.R. Clarke in the _Comparisons of the Standards of
Length_, pp. 280-287; details of the calculations are given by the same
author in his _Geodesy_ (1880), pp. 311 et seq.