Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/147

This page has been proofread, but needs to be validated.

DETERMINATION OF ELLIPTIC ORBITS.

131

Preliminary computations relating to the second and third observations.

The formulæ are entirely analogous to those relating to the first observation, the quantities being distinguished by the proper suffixes.

III

Equations of the first hypothesis.

When the preceding quantities have been computed, their numerical values (or their logarithms, when more convenient for computation,) are to be substituted in the following equations:

Components of ${\mathfrak {S}}_{1}$
$q_{1}=\rho _{1}+({\mathfrak {E}}_{1}.{\mathfrak {F}}_{1})$	$\alpha _{1}=A_{1}\xi _{1}(1+R_{1})\left(q_{1}+{\frac {X_{1}}{\xi _{1}}}-({\mathfrak {E}}_{1}.{\mathfrak {F}}_{1})\right)$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ${\text{III}}_{1}$
$r_{1}^{2}=q_{1}^{2}+p_{1}^{2}$	$\beta _{1}=A_{1}\eta _{1}(1+R_{1})\left(q_{1}+{\frac {Y_{1}}{\eta _{1}}}-({\mathfrak {E}}_{1}.{\mathfrak {F}}_{1})\right)$
$R_{1}={\frac {B_{1}}{r_{1}^{3}}}$	$\gamma _{1}=A_{1}\zeta _{1}(1+R_{1})\left(q_{1}+{\frac {Z_{1}}{\zeta _{1}}}-({\mathfrak {E}}_{1}.{\mathfrak {F}}_{1})\right)$

For control:

s_{1}^{2}=\alpha _{1}^{2}+\beta _{1}^{2}+\gamma _{1}^{2}=A_{1}^{2}(1+R_{1})^{2}r_{1}^{2}

Components of ${\mathfrak {S}}'$
$P'={\frac {3R_{1}q_{1}}{(1+R_{1})r_{1}^{2}}}$	$\alpha '=A_{1}\xi _{1}+A_{1}\xi _{1}R_{1}-P'\alpha _{1}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ${\text{III}}'$
	$\beta '=A_{1}\eta _{1}+A_{1}\eta _{1}R_{1}-P'\beta _{1}$
	$\gamma '=A_{1}\zeta _{1}+A_{1}\zeta _{1}R_{1}-P'\gamma _{1}$

Components of ${\mathfrak {S}}_{2}$
$q_{2}=\rho _{2}+({\mathfrak {E}}_{2}.{\mathfrak {F}}_{2})$	$\alpha _{2}=-\xi _{2}(1-R_{2})\left(q_{2}+{\frac {X_{2}}{\xi _{2}}}-({\mathfrak {E}}_{2}.{\mathfrak {F}}_{2})\right)$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ${\text{III}}_{2}$
$r_{2}^{2}=q_{2}^{2}+p_{2}^{2}$	$\beta _{2}=-\eta _{2}(1-R_{2})\left(q_{2}+{\frac {Y_{2}}{\eta _{2}}}-({\mathfrak {E}}_{2}.{\mathfrak {F}}_{2})\right)$
$R_{2}={\frac {B_{2}}{r_{2}^{3}}}$	$\gamma _{2}=-\zeta _{2}(1+R_{2})\left(q_{2}+{\frac {Z_{2}}{\zeta _{2}}}-({\mathfrak {E}}_{2}.{\mathfrak {F}}_{2})\right)$

For control:

s_{2}^{2}=\alpha _{2}^{2}+\beta _{2}^{2}+\gamma _{2}^{2}=(1-R_{2})^{2}r_{2}^{2}

Components of ${\mathfrak {S}}''$
$P''={\frac {3R_{2}q_{2}}{(1-R_{2})r_{2}^{2}}}$	$\alpha ''=-\xi _{2}+\xi _{2}R_{2}+P''\alpha _{2}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ${\text{III}}''$
	$\beta ''=-\eta _{2}+A_{1}\eta _{2}R_{2}+P''\beta _{2}$
	$\gamma ''=-\zeta _{2}+\zeta _{2}R_{2}+P''\gamma _{2}$

Components of ${\mathfrak {S}}_{3}$
$q_{3}=\rho _{3}+({\mathfrak {E}}_{3}.{\mathfrak {F}}_{3})$	$\alpha _{2}=A_{3}\xi _{3}(1+R_{3})\left(q_{3}+{\frac {X_{3}}{\xi _{3}}}-({\mathfrak {E}}_{3}.{\mathfrak {F}}_{3})\right)$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ${\text{III}}_{3}$
$r_{3}^{2}=q_{3}^{2}+p_{3}^{2}$	$\beta _{3}=A_{3}\eta _{3}(1+R_{3})\left(q_{3}+{\frac {Y_{3}}{\eta _{3}}}-({\mathfrak {E}}_{3}.{\mathfrak {F}}_{3})\right)$
$R_{3}={\frac {B_{3}}{r_{3}^{3}}}$	$\gamma _{3}=A_{3}\zeta _{3}(1+R_{3})\left(q_{3}+{\frac {Z_{3}}{\zeta _{3}}}-({\mathfrak {E}}_{3}.{\mathfrak {F}}_{3})\right)$