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/48

 If the variations of the anomaly $v$ are wanted in seconds, both parts also of $\mathrm{d} v$  must be expressed in this manner, that is, it is necessary to take for $/ c$  the value $3548^{\prime \prime} .188$  given in article 6. If, moreover, $\frac{1}{2} p=q$ is introduced instead of $p,$  the formula will bave the following form:

$$\mathrm{d} v=\frac{k \sqrt{ } 2 q}{r r} \mathrm{~d} t-\frac{3 k t}{r r \sqrt{ }^{2} q} \mathrm{~d} q$$

in which are to be used the constant logarithms

$$\log k \sqrt{ } 2=3.7005215724, \log 3 k \sqrt{ } \frac{1}{8}=3.8766128315$$

Moreover the differentiation of the equation

furnishes

$$r=\frac{p}{2 \cos ^{2} \frac{1}{2} v}$$

$$\frac{\mathrm{d} r}{r}=\frac{\mathrm{d} p}{p}+\tan \frac{1}{2} v \mathrm{~d} v$$

or by expressing $\mathrm{d} v$ by means of $\mathrm{d} t$  and $\mathrm{d} p,$

$$\frac{\mathrm{d} r}{r}=\left(\frac{1}{p}-\frac{3 k t \tan \frac{1}{2} v}{2 r r \sqrt{ } p}\right) \mathrm{d} p+\frac{k \sqrt{ } p \tan \frac{1}{2} v}{r r} \mathrm{~d} t .$$

By substituting for $t$ its value in $v,$  the coefficient of $\mathrm{d} p$  is changed into

$\frac{1}{p}-\frac{3 p \tan ^{2} \frac{1}{2} v}{4 r r}-\frac{p \tan ^{\frac{1}{2}} \frac{1}{2} v}{4 r r}=\frac{1}{r}\left(\frac{1}{8}+\frac{1}{2} \tan ^{2} \frac{1}{2} v-\frac{3}{2} \sin ^{2} \frac{1}{2} v-\frac{1}{2} \sin ^{2} \frac{1}{2} v \tan ^{2} \frac{1}{2} v\right)=\frac{\cos v}{2 r} ;$

but the coefficient of $\mathrm{d} t$ becomes $\frac{k \sin v}{r \sqrt{ } p}.$  From this there results

$$\mathrm{d} r=\frac{1}{2} \cos v \mathrm{~d} p+\frac{k \sin v}{\sqrt{p}} \mathrm{~d} t$$

or if we introduce $q$ for $p$

\mathrm{d} r=\cos v \mathrm{~d} q+\frac{k \sin v}{\sqrt{2} q} \mathrm{~d} t .</