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/54

 26.

Example - Retaining for $e$ and $b$  the same values as in the preceding example, let $t=65.41236: v$  and $r$  are required. Using Briggs’s logarithms we have $\log t.$. . . . 1.8156598

$\log \lambda \hbar b^{-\frac{3}{2}}.$. . 6.9702758

$\log N.$. . . . 8.7859356, whence $N=0.06108514.$ From this it is seen that the equation $N=\lambda e \tan F-\log \tan \left(45^{\circ}+\frac{1}{2} F\right)$  is satisfied by $F=25^{\circ} 24^{\prime} 27^{\prime \prime} .66,$  whence we have, by formula III.,

$\log \tan \frac{1}{2} F.$. . 9.3530120

$\log \tan \frac{1}{2} \psi.$. . . 9.5318179

$\log \tan \frac{1}{2} v.$. 9.8211941, and thus $\frac{1}{2} v=33^{\circ} 31^{\prime} 29^{\prime \prime} .89,$ and $v=$  $67^{\circ} 2^{\prime} 59^{\prime \prime} .78.$  Hence, there follows,

C. $\left.\log \cos \frac{1}{2}(v+\psi) \quad \cdot 0.2137476\right\}$

C. $\left.\log \cos \frac{1}{2}(v-\psi) \cdot 0.0145197\right\}$

$\log \frac{p}{2 e}.$. . . 9.9725868

difference. . . . . . . 0.1992279

$\log \tan \left(45^{\circ}+\frac{1}{3} F\right) \quad.$. 0.1992280

$\log r.$. . . 0.2008541.

27.

If equation IV. is differentiated, considering $u, v, \psi,$ as variable at the same time, there results,

$$\frac{\mathrm{d} u}{u}=\frac{\sin \psi \mathrm{d} v+\sin v \mathrm{~d} \psi}{2 \cos \frac{1}{2}(v-\psi) \cos \frac{1}{2}(v+\psi)}=\frac{r \tan \psi}{p} \mathrm{~d} v+\frac{r \sin v}{p \cos \psi} \mathrm{d} \psi .$$

By differentiating in like manner equation XI, the relation between the differential variations of the quantities $u, \psi, N,$ becomes,

$$\frac{\mathrm{d} N}{\lambda}=\left(\frac{1}{2} e\left(1+\frac{1}{u u}\right)-\frac{1}{u}\right) \mathrm{d} u+\frac{(u u-1) \sin \psi}{2 u \cos ^{2} \psi} \mathrm{d} \psi,$$

or

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