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

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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

     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)

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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)