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/55

 Hence, by eliminating $\mathrm{d} u$ by means of the preceding equation we obtain

or

$$\frac{\mathrm{d} N}{\lambda}=\frac{r r}{b b \tan \psi} \mathrm{d} v+\left(1+\frac{r}{p}\right) \frac{r \sin v}{b \cos \psi} \mathrm{d} \psi$$

$$\begin{aligned} \mathrm{d} v & =\frac{b b \tan \psi}{\lambda r r} \mathrm{~d} N-\left(\frac{b}{r}+\frac{b}{p}\right) \frac{\sin v \tan \psi}{\cos \psi} \mathrm{d} \psi \\ & =\frac{b b \tan \psi}{\lambda r r} \mathrm{~d} N-\left(1+\frac{p}{r}\right) \frac{\sin v}{\sin \psi} \mathrm{d} \psi. \end{aligned}$$

28.

By differentiating equation $\mathrm{X},$ all the quantities $r, b, e, u,$  being regarded as variables, by substituting

$$\mathrm{d} e=\frac{\sin \psi}{\cos ^{2} \psi} \mathrm{d} \psi$$

and eliminating $\mathrm{d} u$ with the help of the equation between $\mathrm{d} N, \mathrm{~d} u, \mathrm{~d} \psi,$  given in the preceding article, there results,

$$\mathrm{d} r=\frac{r}{b} \mathrm{~d} b+\frac{b b e(u u-1)}{2 \lambda u r} \mathrm{~d} N+\frac{b}{2 \cos ^{2} \psi}\left\{\left(u+\frac{1}{u}\right) \sin \psi-\left(u-\frac{1}{u}\right) \sin v\right\} \mathrm{d} \psi \text {. }$$

The coefficient of $\mathrm{d} N$ is transformed, by means of equation VIII., into $\frac{b \sin v}{\lambda \sin \psi};$  but the coefficient of $\mathrm{d} \psi,$  by substituting from equation IV.,

$$u(\sin \psi-\sin v)=\sin (\psi-v), \quad \frac{1}{u}(\sin \psi+\sin v)=\sin (\psi+v)$$

is changed into

$$\frac{b \sin \psi \cos v}{\cos ^{2} \psi}=\frac{p \cos v}{\sin \psi}$$

so that we have

$$\mathrm{d} r=\frac{r}{b} \mathrm{~d} b+\frac{b \sin v}{\lambda \sin \psi} \mathrm{d} N+\frac{p \cos v}{\sin \psi} \mathrm{d} \psi$$

So far, moreover, as $N$ is considered a function of $b$  and $t,$  we have

$$\mathrm{d} N=\frac{N}{t} \mathrm{~d} t-\frac{8}{2} \frac{N}{b} \mathrm{~d} b$$

which value being substituted, we shall have $\mathrm{d} r,$ and also $\mathrm{d} v$  in the preceding article, expressed by means of $\mathrm{d} t, \mathrm{~d} b, \mathrm{~d} \psi.$  Finally, we have here to repeat our