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/53

will come out greater in the ratio ${\textstyle 1:\lambda ,}$ than if Brigg’s logarithms were used. Our example treated according to this method is as follows:-

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

${\textstyle \log \tan {\frac {1}{2}}v.}$ . . . 9.2201009

${\textstyle \log \tan {\frac {1}{2}}F.}$ . . . 8.7519188

$$F=3^{\circ }13^{\prime }58^{\prime \prime }.12$$

${\textstyle \log e.}$ . . . . . 0.1010188

${\textstyle \log \tan F.}$ . . . 9.0543366

9.1553554

C. hyp. ${\textstyle \log \cos {\frac {1}{2}}(v-\psi )=0.01342266}$

e ${\textstyle \tan F=.}$ . . . 0.14300638

C. hyp. ${\textstyle \log \cos {\frac {1}{2}}(v+\psi )=0.12650930}$

hyp. ${\textstyle \log \tan \left(45^{\circ }+{\frac {1}{2}}F\right)=0.11308666}$

Difference . . . . ${\textstyle =0.11308664}$

${\textstyle N=.}$ . . . . 0.02991972

${\textstyle \log N.}$ . . . . . 8.4759575

${\textstyle \log \%.}$ . . . 8.2355814${\textstyle \}}$

${\textstyle \left.{\frac {8}{2}}\log b\cdot ....\quad 0.9030900\right\}}$

Difference . . . . . 7.3324914

${\textstyle \log t.}$ . . . . . 1.1434661

${\textstyle t=\quad 13.91445}$

25.

For the solution of the inverse problem, that of determining the true anomaly and the radius vector from the time, the auxiliary quantity ${\textstyle u}$ or ${\textstyle F}$ must be first derived from ${\textstyle N=\lambda kb^{-{\frac {3}{2}}}t}$ by means of equation XI. The solution of this transcendental equation will be performed by trial, and can be shortened by devices analogous to those we have described in article 11. But we suffer these to pass without further explanation; for it does not seem worth while to elaborate as carefully the precepts for the hyperbolic motion, very rarely perhaps to be exhibited in celestial space, as for the elliptic motion, and besides, all cases that can possibly occur may be solved by another method to be given below. Afterwards ${\textstyle F}$ or ${\textstyle u}$ will be found, thence ${\textstyle v}$ by formula III., and subsequently ${\textstyle r}$ will be determined either by II. or VIII; ${\textstyle v}$ and ${\textstyle r}$ are still more conveniently obtained by means of formulas VI. and VII.; some one of the remaining formulas can be called into use at pleasure, for verifying the calculation.