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/68

 34 might converge too slowly; and therefore it is by no means to be regarded as a defect sof the method about to be explained, that it is specially adapted to those cases in which $E$ or $F$  has not yet increased beyond moderate values.

37.
Let us resume in the elliptic motion the equation between the eccentric anomaly and the time,

$$E-e \sin E=\frac{k t \sqrt{ }(1+\mu)}{a^{\frac{3}{2}}}$$

where we suppose $E$ to be expressed in parts of the radius. Henceforth, we shall leave out the factor $V(1+\mu);$ if a case should occur where it is worth while to take it into account, the symbol $t$  would not express the time itself after perihelion, but this time multiplied by $\sqrt{ }(1+\mu).$  We designate in future by $q$  the perihelion distance, and in the place of $E$  and $\sin E,$  we introduce the quantities

$$E-\sin E, \text { and } E-\frac{1}{10}(E-\sin E)=\frac{9}{10} E+\frac{1}{10} \sin E:$$

the careful reader will readily perceive from what follows, our reason for selecting particularly these expressions. In this way our equation assumes the following form:-

$$(1-e)\left(\frac{9}{10} E+\frac{1}{10} \sin E\right)+\left(\frac{1}{10}+\frac{9}{10} e\right)(E-\sin E)=k t\left(\frac{1-e}{q}\right)^{\frac{9}{2}}$$

As long as $E$ is regarded as a quantity of the first order,

$$\frac{9}{10} E+\frac{1}{10} \sin E=E-\frac{1}{60} E^{3}+\frac{1}{1200} E^{5}-\text { etc. }$$

will be a quantity of the first order, while

$$E-\sin E=\frac{1}{6} E^{3}-\frac{1}{120} E^{5}+\frac{1}{5040} E^{7}-\text { etc. },$$

will be a quantity of the third order. Putting, therefore,

$$\begin{aligned} & \frac{6(E-\sin E)}{\frac{1}{10} E+\frac{1}{10} \sin E}=4 A, \frac{\frac{9}{0} E+\frac{1}{10} \sin E}{2 \sqrt{ } A}=B, \\ & 4 A=E^{2}-\frac{1}{30} E^{4}-\frac{1}{60} E^{6}-\text { etc. } \end{aligned}$$

will be a quantity of the second order, and

$$B=1+\frac{3}{2800} E^{4}-\text { etc. }$$

will differ from unity by a quantity of the fourth order. But hence our equation becomes