Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/159

{{||DETERMINATION OF ELLIPTIC ORBITS.|143}}

VII.

The computations for the test of the uncorrected formula (the first hypothesis) are as follows:

				Number of arc.		Logarithm.
$r_{1}$						0.4282377
$r_{2}$						0.4132937
$r_{3}$						0.4061399
$A_{1}B_{1}r_{1}^{-3}$		⁠$+$⁠		.01174865		8.0699879
$B_{2}r_{2}^{-3}$		$+$		.11980944		9.0784911
$A_{3}B_{3}r_{3}^{-3}$		$+$		.01137670		8.0560162
$N$		$+$		.14293479		9.1551380
$s_{1}$		$+$		1.3308476		0.1241283
$s_{2}$		$+$		2.2796616		0.3578704
$s_{3}$		$+$		1.3417404		0.1276685
$s$		$+$		2.4761248		0.3937725
$s-s_{1}$		$+$		1.1452772		0.0589106
$s-s_{2}$		$+$		0.1964632		9.2932812
$s-s_{3}$		$+$		1.1343844		0.0547602
${R}$		$+$				9.5065898
$p$		$+$				0.4391732
$\tan {\tfrac {1}{2}}(v_{2}-v_{1})$		$+$		15° 48' 10''.82		9.4518296
$\tan {\tfrac {1}{2}}(v_{3}-v_{2})$		$+$		15° 39' 36''.38		9.4476792
$\tan {\tfrac {1}{2}}(v_{3}-v_{1})$		$+$		31° 27' 47''.20		9.7866915
$e\sin {\tfrac {1}{2}}(v_{3}+v_{1})$		$-$				8.7099387
$e\cos {\tfrac {1}{2}}(v_{3}+v_{1})$		$+$				8.7872701
$\tan {\tfrac {1}{2}}(v_{3}+v_{1})$		$-$		–39° 55' 32''.32		9.9226686
$e$		$+$				8.9025438
$\epsilon$		$+$				9.9652259
$a$		$+$				0.4419546
$\tan {\tfrac {1}{2}}v_{1}$		$-$		–35° 41' 39''.75		9.8563809
$\tan {\tfrac {1}{2}}v_{2}$		$-$		–19° 53' 28''.93		9.5384981
$\tan {\tfrac {1}{2}}v_{3}$		$-$		–4° 13' 52''.55		8.8691380
$\tan {\tfrac {1}{2}}E_{1}$		$-$		–33° 33' 0''.17		9.8216068
$\tan {\tfrac {1}{2}}E_{2}$		$-$		–18° 28' 6''.35		9.5237240
$\tan {\tfrac {1}{2}}E_{3}$		$-$		–3° 54' 24''.21		8.8343639
$\sin {\tfrac {1}{2}}E_{1}$		$-$		–67° 6' 0''.34		9.9643473
$\sin {\tfrac {1}{2}}E_{2}$		$-$		–36° 56' 12''.70		9.7788272
$\sin {\tfrac {1}{2}}E_{3}$		$-$		–7° 48' 48''.42		9.1333734
$ea^{\frac {3}{2}}\sin {\tfrac {1}{2}}E_{1}$		$-$		.3387061		9.5298230
$ea^{\frac {3}{2}}\sin {\tfrac {1}{2}}E_{2}$		$-$		.2209545		9.3343029
$ea^{\frac {3}{2}}\sin {\tfrac {1}{2}}E_{3}$		$-$		.0499681		8.6988491
$a^{\frac {3}{2}}(E_{2}-E_{1})$		$+$		2.4226307		0.3842872
$a^{\frac {3}{2}}(E_{3}-E_{2})$		$+$		2.3391145		0.3690515
$\tau _{3{\text{ calc.}}}$		$+$		2.3048791		0.3626482
$\tau _{1{\text{ calc.}}}$		$+$		2.1681461		0.3360885