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/66

 I. In the inverse problem, the determination of the time, that is, from the true anomaly, it is requisite to have recourse to a somewhat indirect method, and to derive $w$ from $v$  by trial. In order to meet this inconvenience, the first series should be treated in the same manner as the second: and since it may be readily perceived that $-v^{\prime}$ is the same function of $v$  as $v^{\prime}$  of $v,$  so that the table for $w^{\prime}$  might answer for $v^{\prime}$  the sign only being changed, nothing more is required than a table for $v^{\prime \prime},$  by which either problem may be solved with equal precision.

Sometimes, undoubtedly, cases may occur, where the eccentricity differs but little from unity, such that the general methods above explained may not appear to afford sufficient precision, not enough at least, to allow the effect of the third and higher powers of $\delta$ in the peculiar method just sketched out, to be safely neglected. Cases of this kind are possible in the hyperbolic motion especially, in which, whether the former methods are chosen or the latter one, an error of several seconds is inevitable, if the common tables, constructed to seven places of figures only, are employed. Although, in truth, such cases rarely occur in practice, something might appear to be wanting if it were not possible in all cases to determine the true anomaly within $0^{\prime \prime} .1,$ or at least $0^{\prime \prime} .2,$  without consulting the larger tables, which would require a reference to books of the rarer sort. We hope, therefore, that it will not seem wholly superfluous to proceed to the exposition of a peculiar method, which we have long had in use, and which will also commend itself on this account, that it is not limited to eccentricities differing but little from unity, but in this respect admits of general application.

36.
Before we proceed to explain this method, it will be proper to observe that the uncertainty of the general methods given above, in orbits approaching the form of the parabola, ceases of itself, when $E$ or $F$  increase to considerable magnitude, which indeed can take place only in large distances from the sun. To show which, we give to

$$\frac{3 \omega e a \sin v}{\lambda r} \cdot 206265^{\prime \prime}$$

the greatest possible error in the ellipse, which we find in article $32, I V,$ the following form,