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/33



6.

Let us proceed now to the comparison of the motion with the time. Putting, as in Art. 1, the space described about the sun in the time $t=\frac{1}{2} g,$ the mass of the moving body $=\mu,$  that of the sun being taken $=1,$  we have $g=k t \sqrt{p} \sqrt{1+\mu}.$  The differential of the space $=\frac{1}{2} r r d v,$  from which there results $k t \sqrt{p} \sqrt{1+\mu}=\int r r \operatorname{d} v,$  this integral being so taken that it will vanish for $t=0.$  This integration must be treated differently for different kinds of conic sections, on which account, we shall now consider each kind separately, beginning with the ELLIPSE.

Since $r$ is determined from $v$  by means of a fraction, the denominator of which consists of two terms, we will remove this inconvenience by the introduction of a new quantity in the place of $v.$  For this purpose we will put $\tan \frac{1}{2} v \sqrt{\frac{1-e}{1+e}}=\tan \frac{1}{2} E,$  by which the last formula for $r$  in the preceding article gives

$$r=\frac{p \cos ^{2} \frac{1}{2} E}{(1+e) \cos ^{2} \frac{1}{2} v}=p\left(\frac{\cos ^{2} \frac{1}{2} E}{1+e}+\frac{\sin ^{2} \frac{1}{2} E}{1-e}\right)=\frac{p}{1-e e}(1-e \cos E)$$

Moreover we have $\frac{\operatorname{d} E}{\cos ^{2} \frac{1}{2} E}=\frac{\operatorname{d} v}{\cos ^{2} \frac{1}{2} v} \sqrt{\frac{1-e}{1+e}},$ and consequently $\operatorname{d} v=\frac{p \operatorname{d} E}{r \sqrt{1-e}};$  hence

$$r r \operatorname{d} v=\frac{r p \operatorname{d} E}{\sqrt{1-e e}}=\frac{p p}{(1-e e)^{\frac{3}{2}}}(1-e \cos E) \operatorname{d} E$$

and integrating,

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

Accordingly, if we place the beginning of the time at the perihelion passage, where $v=0,$ $E=0,$  and thus constant $=0,$  we shall have, by reason of $\frac{p}{1-e e}=a,$

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

In this equation the auxiliary angle $E,$ which is called the eccentric anomaly, must be expressed in parts of the radius. This angle, however, may be retained in degrees, etc., if $e \sin E$ and $\frac{k t \sqrt{1+\mu}}{a^{\frac{3}{2}}}$  are also expressed in the same manner; these quantities will be expressed in seconds of are if they are multiplied by the