Page:Theory of the motion of the heavenly bodies moving about the sun in conic sections- a translation of Gauss's "Theoria motus." With an appendix (IA theoryofmotionof00gaus).pdf/60

 increase to an amount not to be neglected: for this reason formula III, article 8, is less suitable in this case. The quantity $\delta$ may be expressed thus also,

$$\frac{3(a-r)}{r}=\frac{3 e(\cos v+e)}{1-e e}$$

which formula shows still more clearly when the error $(1+\delta) \omega$ may be neglected.

III. In the use of formula X., article 8, for the computation of the true from the mean anomaly, the $\log \sqrt{ } \frac{a}{r}$ is liable to the error $\left(\frac{1}{2}+\frac{1}{2} \delta\right) \omega,$  and so the $\log$  $\sin \frac{1}{2} \varphi \sin E \sqrt{ } \frac{a}{r}$  to that of $\left(\frac{5}{2}+\frac{1}{2} \delta\right) \omega;$  hence the greatest possible error in the determination of the angles $v-E$  or $v$  is

$$\frac{\omega}{\lambda}(7+\delta) \tan \frac{1}{2}(v-E),$$

or expressed in seconds, if seven places of decimals are employed,

$$\left(0^{\prime \prime} .166+0^{\prime \prime} .024 \delta\right) \tan \frac{1}{2}(v-E)$$

When the eccentricity is not great, $\delta$ and $\tan \frac{1}{2}(v-E)$  will be small quantities, on account of which, this method admits of greater accuracy than that which we have considered in I.: the latter, on the other hand, will be preferable when the eccentricity is very great and approaches nearly to unity, where $\delta$  and $\tan \frac{1}{2}(v-E)$  may acquire very considerable values. It will always be easy to decide, by means of our formulas, which of the two methods is to be preferred.

IV. In the determination of the mean anomaly from the eccentric by means of formula XII., article 8, the error of the quantity $e \sin E,$ computed by the help of logarithms, and therefore of the anomaly itself, $M,$  may amount to

$$\frac{3 \omega e \sin E}{\lambda}$$

which limit of error is to be multiplied by $206265^{\prime \prime}$ if wanted expressed in seconds. Hence it is readily inferred, that in the inverse problem where $E$ is to be determined from $M$  by trial, $E$  may be erroneous by the quantity

$$\frac{3 \omega e \sin E}{\lambda} \cdot \frac{\mathrm{d} E}{\mathrm{~d} M} \cdot 206265^{\prime \prime}=\frac{3 \omega e a \sin E}{\lambda r} \cdot 206265^{\prime \prime} \text {, }$$

even if the equation $E-e \sin E=M$ should be satisfied with all the accuracy which the tables admit.