Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/154

XII.

When an approximate orbit is known in advance, we may use it to improve our fundamental equation. The following appears to be the most simple method:

Find the excentric anomalies $E_{1},E_{2},E_{3},$ and the heliocentric distances $r_{1},r_{2},r_{3},$ which belong in the approximate orbit to the times of observation corrected for aberration.

Calculate $B_{1},B_{3},$ as in § I, using these corrected times.

Determine $A_{1},A_{3}$ by the equation

${\frac {A_{1}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right)}{\sin(E_{3}-E_{2})-e\sin E_{3}+e\sin E_{2}}}={\frac {A_{3}\left(1+{\frac {B_{3}}{r_{3}^{3}}}\right)}{\sin(E_{2}-E_{1})-e\sin E_{2}+e\sin E_{1}}}$

in connection with the relation $A_{1}+A_{3}=1.$

Determine $B_{2}$ so as to make

${\frac {A_{1}{\frac {B_{1}}{r_{1}^{3}}}+{\frac {B_{2}}{r_{2}^{3}}}+A_{3}{\frac {B_{3}}{r_{3}^{3}}}}{4\sin {\tfrac {1}{2}}(E_{2}-E_{1})\sin {\tfrac {1}{2}}(E_{3}-E_{2})\sin {\tfrac {1}{2}}(E_{3}-E_{1})}}$

equal to either member of the last equation.

It is not necessary that the times for which $E_{1},E_{2},E_{3},r_{1},r_{2},r_{3},$ are calculated should precisely agree with the times of observation corrected for aberration. Let the former be represented by $t_{1}',t_{2}',t_{3}'$ and the latter by $t_{1}'',t_{2}'',t_{3}'';$ and let

${\begin{aligned}\Delta \log \tau _{1}&=\log(t_{3}''-t_{2}'')-\log(t_{3}'-t_{2}'),\\\Delta \log \tau _{2}&=\log(t_{2}''-t_{1}'')-\log(t_{2}'-t_{1}').\end{aligned}}$

We may find $B_{1},B_{3},A_{1},A_{3},B_{2},$ as above, using $t_{1}',t_{2}',t_{3}',$ and then use $\Delta \log \tau _{1},\Delta \log \tau _{2}$ to correct their values, as in § VIII.

Numerical Example.

To illustrate the numerical computations we have chosen the following example, both on account of the large heliocentric motion, and because Gauss and Oppolzer have treated the same data by their different methods.

The data are taken from the Theoria Motus, § 169, viz.,

Times, 1805, September		5.51336		139.42711		265.39813
Longitudes of Ceres		95° 32' 18''.56		99° 49' 5''.87		118° 5' 28''.85
Latitudes of Ceres		-0° 59' 34''.06		+7° 16' 36''.80		+7° 38' 49''.39
Longitudes of the Earth		342° 54' 56''.00		117° 12' 43''.25		241° 58' 50''.71
Logs of the Sun's distance		0.0031514		9.9929861		0.0056974

The positions of Ceres have been freed from the effects of parallax and aberration.