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 80

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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 80

     III. The values of a, c, and x having been obtained, substitute them in the equations of condition and obtain the residuals, v[subscript]1[/subscript], [subscript]2[/subscript], [subscript]3[/subscript], etc.
     Thus,
     a[subscript]1[/subscript]a + c[subscript]1[/subscript]c + d[subscript]1[/subscript]x - n[subscript]1[/subscript] = -v[subscript]1[/subscript]

     a[subscript]2[/subscript]a + c[subscript]2[/subscript]c + d[subscript]2[/subscript]x - n[subscript]2[/subscript] = -v[subscript]2[/subscript]

     a[subscript]3[/subscript]a + c[subscript]3[/subscript]c + d[subscript]3[/subscript]x - n[subscript]3[/subscript] = -v[subscript]3[/subscript]

          etc.

     Squaring each side of each equation and adding, we get

[aa]a[superscript]2[/superscript] + [cc]c[superscript]2[/superscript] + [dd]x[superscript]2[/superscript] + [nn] 2 [ac]ac + 2 [ad] ax - [an] a + 2 [cd]cx - 2[cn]c - 2 [dn]x = [vv]. (15).

     Multiplying the normal equations (1), (2), (3), by a, c, and x respectively, and adding, we get [aa]a[superscript]2[/superscript] + [cc]c[superscript]2[/superscript] + [dd]x[superscript]2[/superscript] + 2[ac]ac + 2 [ad]ax + 2[cd]cx = [an]a + [cn]c + [dn]x. (16)

Comparing (15) and (16) we have 
[nn] - [an]a - [cn]c - [dn]x = [vv]  (17).
     
     From (1) we have

          a = ([an]/[aa]) - ([ac]/[aa])(c) - ([ad]/[aa])(x).

     Substituting the value of a in (17) we obtain

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III. The values of a, c, and x having been obtained, substitute them in the equations of condition and obtain the residuals, v[subscript]1[/subscript], [subscript]2[/subscript], [subscript]3[/subscript], etc. Thus, a[subscript]1[/subscript]a + c[subscript]1[/subscript]c + d[subscript]1[/subscript]x - n[subscript]1[/subscript] = -v[subscript]1[/subscript] a[subscript]2[/subscript]a + c[subscript]2[/subscript]c + d[subscript]2[/subscript]x - n[subscript]2[/subscript] = -v[subscript]2[/subscript] a[subscript]3[/subscript]a + c[subscript]3[/subscript]c + d[subscript]3[/subscript]x - n[subscript]3[/subscript] = -v[subscript]3[/subscript] etc. Squaring each side of each equation and adding, we get [aa]a[superscript]2[/superscript] + [cc]c[superscript]2[/superscript] + [dd]x[superscript]2[/superscript] + [nn] 2 [ac]ac + 2 [ad] ax - [an] a + 2 [cd]cx - 2[cn]c - 2 [dn]x = [vv]. (15). Multiplying the normal equations (1), (2), (3), by a, c, and x respectively, and adding, we get [aa]a[superscript]2[/superscript] + [cc]c[superscript]2[/superscript] + [dd]x[superscript]2[/superscript] + 2[ac]ac + 2 [ad]ax + 2[cd]cx = [an]a + [cn]c + [dn]x. (16) Comparing (15) and (16) we have [nn] - [an]a - [cn]c - [dn]x = [vv] (17). From (1) we have a = ([an]/[aa]) - ([ac]/[aa])(c) - ([ad]/[aa])(x). Substituting the value of a in (17) we obtain