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/62

 rithms, assuming $e$ and $u$  or $e$  and $F$  to be known exactly, the first part will be liable to the error

$$\frac{5(u u-1) e \omega}{2 u},$$

if it has been computed in the form

$$\frac{\lambda e(u-1)(u+1)}{2 u}$$

or to the error

$$\frac{3(u u+1) e \omega}{2 u}$$

if computed in the form

$$\frac{1}{2} \lambda e u-\frac{\lambda e}{2 u}$$

or to the error $3 e \omega \tan F$ if computed in the form $\lambda e \tan F,$  provided we neglect the error committed in $\log \lambda$  or $\log \frac{1}{2} \lambda.$  In the first case the error can be expressed also by $5 e \omega \tan F,$  in the second by $\frac{3 e \omega}{\cos F},$  whence it is apparent that the error is the least of all in the third case, but will be greater in the first or second, accurding as $u$  or $\frac{1}{u}>2$  or $<2,$  or according as $\pm F^{\prime}>36^{\circ} 52^{\prime}$  or $<36^{\circ} 52^{\prime}.$  But, in any case, the second part of $N$  will be liable to the error $\omega.$

VII. On the other hand, it is evident that if $u$ or $F$  is derived from $N$  by trial, $u$  would be liable to the error

$$(\omega \pm 5 e \omega \tan F) \frac{\mathrm{d} u}{\mathrm{~d} N}$$

or to

$$\left(\omega+\frac{3 e \omega}{\cos F}\right) \frac{\mathrm{d} u}{\mathrm{~d} N}$$

according as the first term in the value of $N$ is used separated into factors, or into terms; $F,$  however, is liable to the error

$$(\omega \pm 3 e \omega \tan F) \frac{\mathrm{d} F}{\mathrm{~d} N} .$$

The upper signs serve after perihelion, the lower before perihelion. Now if $\frac{\mathrm{d} v}{\mathrm{~d} N}$ is substituted here for $\frac{\mathrm{d} u}{\mathrm{~d} N}$  or for $\frac{\mathrm{d} F}{\mathrm{~d} N},$  the effect of this error appears in the determination of $v,$  which therefore will be