Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/155

I.

From the given times we obtain the following values:

		Numbers.		Logarithms.
$t_{2}-t_{1}$		133.91375		2.1268252
$t_{3}-t_{2}$		125.97102		2.1002706
$t_{3}-t_{1}$		259.88477		2.4147809
$A_{1}$		.4847187		9.6854897
$A_{3}$		.5152812		9.7120443
$\tau _{1}$				.3358520
$\tau _{3}$				.3624066
$B_{1}$				9.6692113
$B_{2}$				.3183722
$B_{3}$				9.5623916

Control:

${\begin{aligned}A_{1}B_{1}+B_{2}&+A_{3}B_{3}&=2.4959086\\&{\tfrac {1}{2}}\tau _{1}\tau _{3}&=2.4959081\end{aligned}}$

II.

From the given positions we get:

$\log X_{1}$		9.9835515		$+$			$\log X_{2}$		9.6531725		$-$			$\log X_{3}$		9.6775810		$-$
$\log Y_{1}$		9.4711748		$-$			$\log Y_{2}$		9.9420444		$+$			$\log Y_{3}$		9.515547		$-$
$Z_{1}$		0					$Z_{2}$		0					$Z_{3}$		0
$\log \xi _{1}$		8.9845270		$-$			$\log \xi _{2}$		9.2282738		$-$			$\log \xi _{3}$		9.6690294		$-$
$\log \eta _{1}$		9.9979027		$+$			$\log \eta _{2}$		9.9900800		$+$			$\log \eta _{3}$		9.9416855		$+$
$\log \zeta _{1}$		8.2387150		$-$			$\log \zeta _{2}$		9.1026549		$+$			$\log \zeta _{3}$		9.1240813		$+$
${\mathfrak {E}}_{1}.{\mathfrak {F}}_{1}$		.3874081		$-$			${\mathfrak {E}}_{2}.{\mathfrak {F}}_{2}$		.9314223		$+$			${\mathfrak {E}}_{3}.{\mathfrak {F}}_{3}$		.5599304		$-$
$p_{1}^{2}$		.8645336		$+$			$p_{2}^{2}$		.10006681		$+$			$p_{3}^{2}$		.7130624		$+$

III.

The preceding computations furnish the numerical values for the equations ${\text{III}}_{1},{\text{III'}},{\text{III}}_{2},{\text{III}}'',{\text{III}}_{3},{\text{III}}''',$ which follow. Brackets indicate that logarithms have been substituted for numbers.

We have now to assume some values for the heliocentric distances $r_{1},r_{2},r_{3}.$ A mean proportional between the mean distances of Mars and Jupiter from the Sun suggests itself as a reasonable assumption. In order, however, to test the convergence of the computations, when the assumptions are not happy, we will make the much less probable assumption (actually much farther from the truth) that the heliocentric distances are an arithmetical mean between the distances of Mars and Jupiter. This gives .526 for the logarithm of each of the distances $r_{1},r_{2},r_{3}.$ From these assumed values we compute the first column of numbers in the three following tables: