DIY History | Transcribe | Scholarship at Iowa | Theory of the astronomical transit instrument applied to the portable transit instrument Wuerdemann no.26: a compilation from various authorities, with original observations by Harry Edward Burton, 1903 | Theory of the astronomical transit instrument applied to the portable transit instrument Wuerdemann no. 26: a compilation from various authorities, with original observations by Harry Edward Burton, 1903, Page 77

Theory of the astronomical transit instrument applied to the portable transit instrument Wuerdemann no. 26: a compilation from various authorities, with original observations by Harry Edward Burton, 1903, Page 77

The normal equations are therefore
2.1825 a + 0.2852 c + 1.467 x = -1.983
0.2852 a + 10.9715 c + 3.034 x = +2.146
1.4670 a + 3.0340 c + 5.900 x = -0.19 .
Solution by Gauss's Method.
Let the normal equations be written again in the form
[aa]a + [ac]c + [ad]x = [an] (1)
[ac]a + [cc]c + [cd]x = [cn] (2)
[ad]a + [cd]c + [dd]x = [dn] (3)
From (1) , a = ([an]/[aa]) - (([ac]/[aa])c) - (([ad]/[aa])x) .
Substituting this value of a in (2) and (3) we get
([cc] - (([ac]/[aa])[ac]))c + ([cd] - (([ac]/[aa])[ad]))x = ([cn] - (([ac]/[aa])[an])) (4)
and
([cd] - (([ad]/[aa])[ac]))c + ([dd] - (([ad]/[aa])[ad]))x = ([dn] - (([ad]/[aa])[an])) (5)
Now for the sake of abbreviation, let
[cc] - (([ac]/[aa])[ac]) = [cc1],
[cd] - (([ac]/[aa])[ad]) = [cd1],
[cn] - (([ac]/[aa])[an]) = [cn1],
[dd] - (([ad]/[aa])[ad]) = [dd1],
[dn] - (([ad]/[aa])[an]) = [dn1] . } (6)

The normal equations are therefore
2.1825 a + 0.2852 c + 1.467 x = -1.983
0.2852 a + 10.9715 c + 3.034 x = +2.146
1.4670 a + 3.0340 c + 5.900 x = -0.19 .
Solution by Gauss's Method.
Let the normal equations be written again in the form
[aa]a + [ac]c + [ad]x = [an] (1)
[ac]a + [cc]c + [cd]x = [cn] (2)
[ad]a + [cd]c + [dd]x = [dn] (3)
From (1) , a = ([an]/[aa]) - (([ac]/[aa])c) - (([ad]/[aa])x) .
Substituting this value of a in (2) and (3) we get
([cc] - (([ac]/[aa])[ac]))c + ([cd] - (([ac]/[aa])[ad]))x = ([cn] - (([ac]/[aa])[an])) (4)
and
([cd] - (([ad]/[aa])[ac]))c + ([dd] - (([ad]/[aa])[ad]))x = ([dn] - (([ad]/[aa])[an])) (5)
Now for the sake of abbreviation, let
[cc] - (([ac]/[aa])[ac]) = [cc1],
[cd] - (([ac]/[aa])[ad]) = [cd1],
[cn] - (([ac]/[aa])[an]) = [cn1],
[dd] - (([ad]/[aa])[ad]) = [dd1],
[dn] - (([ad]/[aa])[an]) = [dn1] . } (6)