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/64

 33.

The methods above treated, both for the determination of the true anomaly from the time and for the determination of the time from the true anomaly, ${ }^{*}$ do not admit of all the precision that might be required in those conic sections of which the eccentricity differs but little from unity, that is, in ellipses and hyperbolas which approach very near to the parabola; indeed, unavoidable errors, increasing as the orbit tends to resemble the parabola, may at length exceed all limits. Larger tables, constructed to more than seven figures would undoubtedly diminish this uncertainty, but they would not remove it, nor would they prevent its surpassing all limits as soon as the orbit approached too near the parabola. Moreover, the methods given above become in this case very troublesome, since a part of them require the use of indirect trials frequently repeated, of which the tediousness is even greater if we work with the larger tables. It certainly, therefore, will not be superfluous, to furnish a peculiar method by means of which the uncertainty in this case may be avoided, and sufficient precision may be obtained with the help of the common tables.

34.
The common method, by which it is usual to remedy these inconveniences, rests upon the following principles. In the ellipse or hyperbola of which $e$ is the eccentricity, $p$  the semi-parameter, and therefore the perihelion distance

$$\frac{p}{1+e}=q,$$

let the true anomaly $v$ correspond to the time $t$  after the perihclion; in the parabola of which the semi-parameter $=2 q,$  or the perihelion distance $=q,$  let the true anomaly $w$  correspond to the same time, supposing in each case the mass $\mu$  to be either neglected or equal. It is evident that we then have


 * Since the time contains the factor $a^{\frac{3}{2}}$ or $b^{\frac{3}{2}},$  the greater the values of $a=\frac{p}{1-e e},$  or $b=\frac{p}{e^{2}-1},$  the more the error in $M$  or $N$  will be increased.