Page:Practical astronomy (1902, John Wiley & Sons).djvu/22

Subtracting from (4) we have

(6)

$n(t-t_{p})=\theta ^{\prime }-l_{p}-2e\sin \theta +{\tfrac {3}{4}}e^{2}\sin 2\theta$,

which since

(7)

$\theta ^{\prime }-l_{p}=\theta$

reduces to

(8)

$n(t-t_{p})=\left(\theta ^{\prime }-l_{p}\right)-2e\sin \left(\theta ^{\prime }-l_{p}\right)+{\tfrac {3}{4}}e^{2}\sin 2\left(\theta ^{\prime }-l_{p}\right)$.

Transposing $l_{p}$, we have

(9)

$n(t-t_{p})+l_{p}=l_{m}=\theta ^{\prime }-2e\sin \left(\theta ^{\prime }-l_{p}\right)+{\tfrac {3}{4}}e^{2}\sin 2\left(\theta ^{\prime }-l_{p}\right)$,

in which $l_{m}$ is the longitude of the mean place of the earth at the time $t$, referred to the same origin.

Let $L$ be the longitude of the earth's mean place at the epoch, also referred to the same origin, and $T$ any interval of time before or after this epoch.

Then will

(10)

$l_{m}=L+nT$,

and we have

(11)

$L+nT=\theta ^{\prime }-2e\sin \left(\theta ^{\prime }-l_{p}\right)+{\tfrac {3}{4}}e^{2}\sin 2\left(\theta ^{\prime }-l_{p}\right)$.

To find the values of the four unknown quantities, $L$, $n$, $e$, and $l_{p}$, take four observations of R. A. and declination at different times, and having reduced the declination to its geocentric value by correcting for refraction and parallax, find the corresponding longitudes (Art. 180, Young).

Each longitude is necessarily referred to the true equinox of its own date. Reduce each to the mean equinox of the epoch by correcting for aberration, nutation, precession, and perturbations, add 180°, and the results will be the longitudes of the true place of the earth referred to a common point—the mean equinox of the epoch.

They will therefore be the values of $\theta ^{\prime }$ corresponding to the values of $T$ in the following equations, the solution of which will give $L$, $n$, $e$, and $l_{p}$.

(12)

$\left.{\begin{aligned}L+nT_{1}=\theta _{1}^{\prime }-2e\sin \left(\theta ^{\prime }-l_{p}\right)\\L+nT_{2}=\theta _{2}^{\prime }-2e\sin \left(\theta ^{\prime }-l_{p}\right)\\L+nT_{3}=\theta _{3}^{\prime }-2e\sin \left(\theta ^{\prime }-l_{p}\right)\\L+nT_{4}=\theta _{4}^{\prime }-2e\sin \left(\theta ^{\prime }-l_{p}\right)\end{aligned}}\right\rbrace$.