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/61

 The true anomaly therefore computed from the mean may be incorrect in two ways, if we consider the mean as given accurately; first, on account of the error committed in the computation of $v$ from $E,$  which, as we have seen, is of slight importance; second, because the value of the eccentric anomaly itself may be erroneous. The effect of the latter cause will be expressed by the product of the error committed in $E$ into $\frac{\mathrm{d} v}{\mathrm{~d} E},$  which product becomes

$$\frac{3 \omega e \sin E}{\lambda} \cdot \frac{\mathrm{d} v}{\mathrm{~d} M} \cdot 206265^{\prime \prime}=\frac{3 \omega e a \sin v}{\lambda r} \cdot 206265^{\prime \prime}=\left(\frac{e \sin v+\frac{1}{2} e e \sin 2 v}{1-e e}\right) 0^{\prime \prime} .0712$$

if seven places of decimals are used. This error, always small for small values of $e,$ may become very large when $e$  differs but little from unity, as is shown by the following table, which exhibits the maximum value of the preceding expression for certain values of $e.$

V. In the hyperbolic motion, if $v$ is determined by means of formula III., article 21, from $F$  and $\psi$  accurately known, the error may amount to

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

but if it is computed by means of the formula

$$\tan \frac{1}{2} v=\frac{(u-1) \tan \frac{1}{2} \psi}{u+1}$$

$u$ and $\psi$  being known precisely, the limit of the error will be one third greater, that is,

$$\frac{4 \omega \sin v}{\lambda} \cdot 206265^{\prime \prime}=0^{\prime \prime} .09 \sin v$$

for seven places.

VI. If the quantity

$$\frac{\lambda k t}{b^{\frac{3}{2}}}=N$$

is computed by means of formula XI, article 22, with the aid of Briggs’s loga-