Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/145

and

$s_{1}-s_{2}+s_{3}={\frac {(s_{1}+s_{4}+s_{3})(-s_{1}+s_{4}+s_{3})(s_{1}-s_{4}+s_{3})(s_{1}+s_{4}-s_{3})}{(s_{1}+s_{2}+s_{3})(-s_{1}+s_{2}+s_{3})(s_{1}+s_{2}-s_{3})}}\cdot$

(39)

The numerical determination of this value of $s_{1}-s_{2}+s_{3}$ is critical only to the first degree.

The eccentricity and the true anomalies may be determined in the same way as in the correction of the formula. The position of the orbit in space may be derived from the following considerations. The vector $-{\mathfrak {S}}_{2}$ is directed from the sun toward the second position of the body; the vector ${\mathfrak {S}}_{4}$ from the first to the third position. If we set

${\mathfrak {S}}_{5}={\mathfrak {S}}_{4}-{\frac {{\mathfrak {S}}_{4}.{\mathfrak {S}}_{2}}{s_{2}^{2}}}{\mathfrak {S}}_{2},$

(40)

the vector ${\mathfrak {S}}_{5}$ will be in the plane of the orbit, perpendicular to $-{\mathfrak {S}}_{2}$ and on the side toward which anomalies increase. If we write $s_{5}$ for the length of ${\mathfrak {S}}_{5},$

$-{\frac {{\mathfrak {S}}_{2}}{s_{2}}}$⁠${\frac {{\mathfrak {S}}_{5}}{s_{5}}}$

will be unit vectors. Let ${\mathfrak {J}}$ and ${\mathfrak {J}}'$ be unit vectors determining the position of the orbit, ${\mathfrak {J}}$ being drawn from the sun toward the perihelion, and ${\mathfrak {J}}'$ at right angles to ${\mathfrak {J}},$ in the plane of the orbit, and on the side toward which anomalies increase. Then

${\mathfrak {J}}=-\cos v_{2}{\frac {{\mathfrak {S}}_{2}}{s_{2}}}-\sin v_{2}{\frac {{\mathfrak {S}}_{5}}{s_{5}}},$

(41)

${\mathfrak {J}}'=-\sin v_{2}{\frac {{\mathfrak {S}}_{2}}{s_{2}}}+\cos v_{2}{\frac {{\mathfrak {S}}_{5}}{s_{5}}}\cdot$

(42)

The time of perihelion passage (${\text{T}}$) may be determined from any one of the observations by the equation

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

(43)

the eccentric anomaly $E$ being calculated from the true anomaly $v.$ The interval $t-T$ in this equation is to be measured in days. A better value of $T$ may be found by averaging the three values given by the separate observations, with such weights as the circumstances may suggest. But any considerable differences in the three values of $T$ would indicate the necessity of a second correction of the formula, and furnish the basis for it.

For the calculation of an ephemeris we have

${\mathfrak {R}}=-ae{\mathfrak {J}}+\cos E\,a{\mathfrak {J}}+\sin E\,b{\mathfrak {J}}'$

(44)

in connection with the preceding equation.

Sometimes it may be worth while to make the calculations for the correction of the formula in the slightly longer form indicated for