Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/146

the determination of the orbit. This will be the case when we wish simultaneously to correct the formula for its theoretical imperfection, and to correct the observations by comparison with others not too remote. The rough approximation to the orbit given by the uncorrected formula may be sufficient for this purpose. In fact, for observations separated by very small intervals, the imperfection of the uncorrected formula will be likely to affect the orbit less than the errors of the observations.

The computer may prefer to determine the orbit from the first and third heliocentric positions with their times. This process, which has certain advantages, is perhaps a little longer than that here given, and does not lend itself quite so readily to successive improvements of the hypothesis. When it is desired to derive an improved hypothesis from an orbit thus determined, the formulæ in § XII of the summary may be used.

SUMMARY OF FORMULÆ

WITH DIRECTIONS FOR USE.

(For the case in which an approximate orbit is known in advance, see XII.)

I.

Preliminary computations relating to the intervals of time.

$t_{1},t_{2},t_{3}=$ times of the observations in days,

$\log k=8.2355814$ (after Gauss)

$\tau _{1}=k(t_{3}-t_{2})$ $\tau _{3}=k(t_{2}-t_{1})$

$A_{1}={\frac {t_{3}-t_{2}}{t_{3}-t_{1}}}$$A_{3}={\frac {t_{2}-t_{1}}{t_{3}-t_{1}}}$

$B_{1}={\frac {\tau _{1}^{2}+\tau _{1}\tau _{3}+\tau _{3}^{2}}{12}}$$B_{2}={\frac {\tau _{1}^{2}+3\tau _{1}\tau _{3}+\tau _{3}^{2}}{12}}$$B_{3}={\frac {\tau _{1}^{2}+\tau _{1}\tau _{3}-\tau _{3}^{2}}{12}}$

For control: $A_{1}B_{1}+B_{2}+A_{3}B_{3}={\tfrac {1}{2}}\tau _{1}\tau _{3}.$

II.

Preliminary computations relating to the first observation.

$X_{1},Y_{1},Z_{1}$ (components of ${\mathfrak {E}}_{1}$) $=$ the heliocentric coordinates of the earth, increased by the geocentric coordinates of the observatory.

$\xi _{1},\eta _{1},\zeta _{1}$ (components of ${\mathfrak {F}}_{1}$) $=$ the direction-cosines of the observed position, corrected for the aberration of the fixed stars.

${\mathfrak {E}}_{1}^{2}=X_{1}^{2}+Y_{1}^{2}+Z_{1}^{2}$$({\mathfrak {E}}_{1}.{\mathfrak {F}}_{1})=X_{1}\xi _{1}+Y_{1}\eta _{1}+Z_{1}\zeta _{1}$$p_{1}^{2}={\mathfrak {E}}_{1}^{2}-({\mathfrak {E}}_{1}.{\mathfrak {F}}_{1})^{2}$