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/69

 $$B\left(2(1-e) A^{\frac{1}{2}}+\frac{2}{15}(1+9 e) A^{\frac{1}{2}}\right)=k t\left(\frac{1-e}{q}\right)^{\frac{3}{2}} \ldots. \cdot[1]$$

By means of the common trigonometrical tables, $\frac{9}{10} E+\frac{1}{10} \sin E$ may be computed with sufficient accuracy, but not $E-\sin F,$  when $E$  is a small angle; in this way therefore it would not be possible to determine correctly enough the quantities $\dot{A}$  and $B.$  A remedy for this difficulty would be furnished by an appropriate table, from which we could take out with the argument $E,$  either $B$  or the logarithm of $B;$  the means necessary to the construction of such a table will readily present themselves to any one even moderately versed in analysis. By the aid of the equation

$$\frac{9 E+\sin E}{20 B}=\sqrt{ } A$$

$\checkmark A$ can be determined, and hence $t$  by formula [1] with all desirable precision.

The following is a specimen of such a table, which will show the slow increase of $\log B;$ it would be superfluous to take the trouble to extend this table, for further on we are about to describe tables of a much more convenient form.

38.

It will not be useless to illustrate by an example what has been given in the preceding article. Let the proposed true anomaly $=100^{\circ},$ the eccentricity $=0.96764567, \log q=9.7656500.$  The following is the calculation for $E, B,$  $A,$  and $t$  :

UTF8 mj 一

$\log \tan \frac{1}{2} v.$. . . 0.0761865

$\log \vee^{\frac{1-e}{1+e}}.$. . . 9.1079927.

$\log \tan \frac{1}{8} E.$. . 9.1841792, whence $\frac{1}{2} E=8^{\circ} 41^{\prime} 19^{\prime \prime} .32,$ and $E=$