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/63

 $$\frac{b b \tan \psi(1 \pm 3 e \tan F) \omega}{\lambda r r} \text { or } \frac{b b \tan \psi(1+3 e \sec F) \omega}{\lambda r r},$$

if the auxiliary quantity $u$ has been employed; on the other hand, if $F$  has been used, this effect becomes,

$$\frac{b b \tan \psi(1 \pm 3 e \tan F) \omega}{\lambda r r}=\frac{\omega}{\lambda}\left\{\frac{(1+e \cos v)^{2}}{\tan ^{8} \psi} \pm \frac{3 e \sin v(1+e \cos v)}{\tan ^{2} \psi}\right\} .$$

If the error is to be expressed in seconds, it is necessary to apply the factor 206265 &quot;. It is evident that this error can only be considerable when $\psi$ is a small angle, or $e$  a little greater than 1. The following are the greatest values of this third expression, for certain values of $e,$ if seven places of decimals are employed:

To this error arising from the erroneous value of $F$ or $u$  it is necessary to apply the error determined in $\mathrm{V}.$  in order to have the total uncertainty of $v.$

VIII. If the equation XI., article 22, is solved by the use of hyperbolic logarithms, $F$ being employed as an auxiliary quantity, the effect of the possible error in this operation in the determination of $v,$  is found by similar reasoning to be,

$$\frac{(1+e \cos v)^{2} \omega^{\prime}}{\tan ^{8} \psi} \pm \frac{3 e \sin v(1+e \cos v) \omega}{\lambda \tan ^{2} \psi}$$

where by $\omega^{\prime}$ we denote the greatest uncertainty in the tables of hyperbolic logarithms. The second part of this expression is identical with the second part of the expression given in VII.; but the first part in the latter is less than the first in the former, in the ratio $\lambda \omega^{\prime}: \omega,$ that is, in the ratio $1: 23,$  if it be admissible to assume that the table of Ursin is everywhere exact to eight figures, or

$$\omega^{\prime}=0.000000005$$