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/59

 32.

Let us proceed now to the application of these principles to the most useful of the operations above explained.

I. If $\varphi$ and $E$  are supposed to be exactly given in using the formula VII., article 8, for computing the true anomaly from the eccentric anomaly in the elliptic motion, then in $\log \tan \left(45^{\circ}-\frac{1}{2} \varphi\right)$  and $\log \tan \frac{1}{3} E,$  the error $\omega$  may be committed, and thus in the difference $=\log \tan \frac{1}{2} v,$  the error $2 \omega;$  therefore the greatest error in the determination of the angle $\frac{1}{2} v$  will be

$$\frac{3 \omega \mathrm{d} \frac{1}{2} v}{\mathrm{~d} \log \tan \frac{1}{2} v}=\frac{3 \omega \sin v}{2 \lambda}$$

 denoting the modulus of the logarithms used in this calculation. The error, therefore, to which the true anomaly $v$ is liable, expressed in seconds, becomes 

$$\frac{3 \omega \sin v}{\lambda} 206265=0^{\prime \prime} .0712 \sin v,$$

if Brigg’s logarithms to seven places of decimals are employed, so that we may be assured of the value of $v$ within $0^{\prime \prime} .07;$  if smaller tables to five places only, are used, the error may amount to $7^{\prime \prime} .12.$

II. If $e \cos E$ is computed by means of logarithms, an error may be committed. to the extent of

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

therefore the quantity

$$1-e \cos E, \text { or } \frac{r}{a},$$

will be liable to the same error. In computing, accordingly, the logarithm of this quantity, the error may amount to $(1+\delta) \omega,$ denoting by $\delta$  the quantity

$$\frac{3 e \cos E}{1-e \cos E}$$

taken positively: the possible error in $\log r$ goes up to the same limit, $\log a$  being assumed to be correctly given. If the eccentricity is small, the quantity $\delta$ is always confined within narrow limits; but when $e$  differs but little from 1, $1-e \cos E$  remains very small as long as $E$  is small; consequently, $\delta$  may