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/39

 that the changes of the logarithm of the sine from $\varepsilon$ to $\varepsilon+x,$  and the changes of the logarithm of the number from $e \sin \varepsilon$  to $e \sin (\varepsilon+x),$  can be regarded as uniform, we may evidently put

$$e \sin (\varepsilon+x)=e \sin \varepsilon \pm \frac{\lambda x}{\mu},$$

the upper sign belonging to the first and fourth quadrants, and the lower to the second and third. Whence, since

$\displaystyle \varepsilon+x=M+e \sin (\varepsilon+x)$, we have $\displaystyle x=\frac{\mu}{\mu+\lambda}(M+e \sin \varepsilon-\varepsilon),$

and the correct value of $E,$ or

$$\varepsilon+x=M+e \sin \varepsilon \pm \frac{\lambda}{\mu+\lambda}(M+e \sin \varepsilon-\varepsilon)$$

the signs being determined by the above-mentioned condition.

Finally, it is readily perceived that we have, without regard to the signs, $\mu: \lambda=1: e \cos \varepsilon,$ and therefore always $\mu>\lambda,$  whence we infer that in the first and last quadrant $M+e \sin \varepsilon$  lies between $\varepsilon$  and $\varepsilon+x,$  and in the second and third, $\varepsilon+x$  between $\varepsilon$  and $M+e \sin \varepsilon,$  which rule dispenses with paying attention to the signs. If the assumed value $\varepsilon$ differs too much from the truth to render the foregoing considerations admissible, at least a much more suitable value will be found by this method, with which the same operation can be repeated, once, or several times if it should appear necessary. It is very apparent, that if the difference of the first value $\varepsilon$ from the truth is regarded as a quantity of the first order, the error of the new value would be referred to the second order, and if the operation were further repeated, it would be reduced to the fourth order, the eighth order, etc. Moreover, the less the eccentricity, the more rapidly will the successive corrections converge.

12.

The approximate value of $E,$ with which to begin the calculation, will, in most cases, be obvious enough, particularly where the problem is to be solved for several values of $M$  of which some have been already found. In the absence of other helps, it is at least evident that $E$ must fall between $M$  and $M \pm e,$  (the eccentricity $e$  being expressed in seconds, and the upper sign being used in the