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/44



16.
The radius vector $r$ is not fully determined by $v$  and $\varphi,$  or by $M$  and $\varphi,$  but depends, besides these, upon $p$  or $a;$  its differential, therefore, will consist of three parts. By differentiating equation II. of article 8, we obtain

$$\frac{\mathrm{d} r}{r}=\frac{\mathrm{d} p}{p}+\frac{e \sin v}{1+e \cos v} \mathrm{~d} v-\frac{\cos \varphi \cos v}{1+e \cos v} \mathrm{~d} \varphi .$$

By putting here

$$\frac{\mathrm{d} p}{p}=\frac{\mathrm{d} \alpha}{\alpha}-2 \tan \varphi \mathrm{d} \varphi$$

(which follows from the differentiation of equation I.), and expressing, in conformity with the preceding article, $\mathrm{d} v$ by means of $\mathrm{d} M$  and $\mathrm{d} \varphi,$  we have, after making the proper reductions,

$$\begin{aligned} & \frac{\mathrm{d} r}{r}=\frac{\mathrm{d} a}{a}+\frac{a}{r} \tan \varphi \sin v \mathrm{~d} M-\frac{a}{r} \cos \varphi \cos v \mathrm{~d} \varphi, \\ & \mathrm{d} r=\frac{r}{a} \mathrm{~d} a+a \tan \varphi \sin v \mathrm{~d} M-a \cos \varphi \cos v \mathrm{~d} \varphi. \end{aligned}$$

Finally, these formulas, as well as those which we developed in the preceding article, rest upon the supposition that $v, \varphi,$ and $M,$  or rather $\mathrm{d} v, \mathrm{~d} \varphi,$  and $\mathrm{d} M,$  are expressed in parts of the radius. If, therefore, we choose to express the variations of the angles $v, \varphi,$ and $M,$  in seconds, we must either divide those parts of the formulas which contain $\mathrm{d} v, \mathrm{~d} \varphi,$  or $\mathrm{d} M,$  by 206264.8, or multiply those which contain $\mathrm{d} r, \mathrm{~d} p, \mathrm{~d} a,$  by the same number. Consequently, the formulas of the preceding article, which in this respect are homogeneous, will require no change.

17.
It will be satisfactory to add a few words concerning the investigation of the greatest equation of the centre. In the first place, it is evident in itself that the difference between the eccentric and mean anomaly is a maximum for $E=90^{\circ},$ where it becomes $=e$  (expressed in degrees, etc.); the radius vector at this point $=a,$  whence $v=90^{\circ}+\varphi,$  and thus the whole equation of the centre $=\varphi+c,$