Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/141

But a little examination will show that the coefficients of ${\displaystyle }$ in these equations will not generally have very different values from the coefficient of the same quantity in equation (24). We may therefore write with sufficient accuracy

$\Delta q_{1}=[\Delta q_{1}]+\Delta q_{2}-[\Delta q_{2}],$$\Delta q_{3}=[\Delta q_{3}]+\Delta q_{2}-[\Delta q_{2}],$

(28)

where $[\Delta q_{1}],[\Delta q_{2}],[\Delta q_{3}]$ denote values obtained from equations (20).

In making successive corrections of the distances $q_{1},q_{2},q_{3},$ it will not be necessary to recalculate the values of ${\mathfrak {S}}',{\mathfrak {S}}'',{\mathfrak {S}}''',$ when these have been calculated from fairly good values of $q_{1},q_{2},q_{3}.$ But when, as is generally the case, the first assumption is only a rude guess, the values of ${\mathfrak {S}}',{\mathfrak {S}}'',{\mathfrak {S}}'''$ should be recalculated after one or two corrections of $q_{1},q_{2},q_{3}.$ To get the best results when we do not recalculate ${\mathfrak {S}}',{\mathfrak {S}}'',{\mathfrak {S}}''',$ we may proceed as follows: Let ${\mathfrak {S}}',{\mathfrak {S}}'',{\mathfrak {S}}'''$ denote the values which have been calculated; $Dq_{1},Dq_{2},Dq_{3},$ respectively, the sum of the corrections of each of the quantities $q_{1},q_{2},q_{3},$ which have been made since the calculation of ${\mathfrak {S}}',{\mathfrak {S}}'',{\mathfrak {S}}''';\,{\mathfrak {S}}$ the residual after all the corrections of $q_{1},q_{2},q_{3},$ which have been made; and $\Delta q_{1},\Delta q_{2},\Delta q_{3}$ the remaining corrections which we are seeking. We have, then, very nearly

$-{\mathfrak {S}}=\{{\mathfrak {S}}'+{\mathfrak {T}}'(Dq_{1}+{\tfrac {1}{2}}\Delta q_{1})\}\Delta q_{1}+\{{\mathfrak {S}}''+{\mathfrak {T}}''(Dq_{2}+{\tfrac {1}{2}}\Delta q_{2})\}\Delta q_{2})$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ (29)
$+\{{\mathfrak {S}}'''+{\mathfrak {T}}'''(Dq_{3}+{\tfrac {1}{2}}\Delta q_{3})\}\Delta q_{3}.$

The same considerations which we applied to equation (21) enable us to simplify this equation also, and to write with a fair degree of accuracy

$-({\mathfrak {S}}{\mathfrak {S}}'''{\mathfrak {S}}')=\{({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')+{\tfrac {6}{5}}({\mathfrak {T}}''{\mathfrak {S}}'''{\mathfrak {S}}')(Dq_{2}+{\tfrac {1}{2}}\Delta q_{2})\}\Delta q_{2},$

(30)

$\Delta q_{1}=[\Delta q_{1}]+\Delta q_{2}-[\Delta q_{2}],$$\Delta q_{3}=[\Delta q_{3}]+\Delta q_{2}-[\Delta q_{2}],$

(31)

where

$[\Delta q_{1}]=-{\frac {({\mathfrak {S}}{\mathfrak {S}}''{\mathfrak {S}}''')}{({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')}},$$[\Delta q_{2}]=-{\frac {({\mathfrak {S}}{\mathfrak {S}}'''{\mathfrak {S}}')}{({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')}},$$[\Delta q_{3}]=-{\frac {({\mathfrak {S}}{\mathfrak {S}}'{\mathfrak {S}}'')}{({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')}}\cdot$

(32)

Correction of the Fundamental Equation.

When we have thus determined, by the numerical solution of our fundamental equation, approximate values of the three positions of the body, it will always be possible to apply a small numerical correction to the equation, so as to make it agree exactly with the laws of elliptic motion in a fictitious case differing but little from the actual. After such a correction the equation will evidently apply to the actual case with a much higher degree of approximation.

There is room for great diversity in the application of this principle. The method which appears to the writer the most simple and direct is the following, in which the correction of the intervals for aberration