Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/153

Direction-cosines of semi-minor axis.
$\lambda$	$=-{\frac {\sin v_{2}}{s_{2}}}\alpha _{2}+{\frac {\cos v_{2}}{s_{5}}}\alpha _{5}$
$\mu$	$=-{\frac {\sin v_{2}}{s_{2}}}\beta _{2}+{\frac {\cos v_{2}}{s_{5}}}\beta _{5}$
$\nu$	$=-{\frac {\sin v_{2}}{s_{2}}}\gamma _{2}+{\frac {\cos v_{2}}{s_{5}}}\gamma _{5}$

Components of the semi-axes.
$a_{x}=al$⁠	$a_{y}=am$⁠	$a_{z}=an$
$b_{x}=b\lambda$	$b_{y}=b\mu$	$b_{z}=b\nu$

X.

Time of perihelion passage.

	Corrections for aberration.
$\tan {\tfrac {1}{2}}E_{1}=\epsilon \tan {\tfrac {1}{2}}v_{1}$⁠	$\delta t_{1}=-.0057613\rho _{1}$
$\tan {\tfrac {1}{2}}E_{2}=\epsilon \tan {\tfrac {1}{2}}v_{2}$⁠	$\delta t_{2}=-.0057613\rho _{2}$
$\tan {\tfrac {1}{2}}E_{3}=\epsilon \tan {\tfrac {1}{2}}v_{3}$⁠	$\delta t_{3}=-.0057613\rho _{3}$
$\log .0057613=7.76052$
$t_{1}+\delta t_{1}-T=k^{-1}a^{\frac {3}{2}}(E_{1}-e\sin E_{1})$
$t_{2}+\delta t_{2}-T=k^{-1}a^{\frac {3}{2}}(E_{2}-e\sin E_{2})$
$t_{3}+\delta t_{3}-T=k^{-1}a^{\frac {3}{2}}(E_{3}-e\sin E_{3})$

The threefold determination of $T$ affords a control of the exactness of the solution of the problem. If the discrepancies in the values of $T$ are such as to require another correction of the formulæ (a third hypothesis), this may be based on the equations

$\Delta \log \tau _{1}=M{\frac {T_{(3)}-T_{(2)}}{t_{3}-t_{2}}}$⁠$\Delta \log \tau _{2}=M{\frac {T_{(2)}-T_{(1)}}{t_{2}-t_{1}}}$

where $T_{(1)},T_{(2)},T_{(3)}$ denote respectively the values obtained from the first, second, and third observations, and $M$ the modulus of common logarithms.

XI.

For an ephemeris.

${\frac {k}{a^{\frac {3}{2}}}}(t-T)=E-e\sin E$

Heliocentric coordinates. (Components of ${\mathfrak {R}}$.)

$x=-ea_{x}+a_{x}\cos E+b_{x}\sin E$

$y=-ea_{y}+a_{y}\cos E+b_{y}\sin E$

$z=-ea_{z}+a_{z}\cos E+b_{z}\sin E$

These equations are completely controlled by the agreement of the computed and observed positions and the following relations between the constants:

$a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z}=0$⁠$a_{x}^{2}+a_{y}^{2}+a_{z}^{2}=a^{2}$⁠$b_{x}^{2}+b_{y}^{2}+b_{z}^{2}=(1-e)^{2}a^{2}$