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/40

 first and second quadrants, the lower in the third and fourth), wherefore, either $M,$ or its value increased or diminished by any estimate whatever, can be taken for the first value of $E.$  It is hardly necessary to observe, that the first calculation, when it is commenced with a value having no pretension to accuracy, does not require to be strictly exact, and that the smaller tables are abundantly sufficient. Moreover, for the sake of convenience, the values selected for $\varepsilon$  should be such that their sines can be taken from the tables without interpolation; as, for example, values to minutes or exact tens of seconds, according as the tables used proceed by differences of minutes or tens of seconds. Every one will be able to determine without assistance the modifications these precepts undergo if the angles are expressed according to the new decimal division.

13.

Example. Let the eccentricity be the same as in article 10. $M=332^{\circ} 28^{\prime} 54^{\prime \prime}.77.$ There the $\log e$  in seconds is $4.7041513,$  therefore $e=50600^{\prime \prime}=14^{\circ} 3^{\prime} 20^{\prime \prime}.$  Now since $E$  here must be less than $M,$  let us in the first calculation put $\varepsilon=326^{\circ},$  then we have by the smaller tables

$$ \begin{array}{ll} \log \sin \varepsilon .\quad.\quad.\quad.\quad. &9.74756 n, \\ \log e \text { in seconds } \quad.\quad.& 4.70415 \\ \hline & 4.45171 n; \end{array} \begin{array}{c} \scriptscriptstyle \text { Change for } 1^{\prime} \ldots 19 \text {, whence } \lambda=0.32.\\ \\ \\ \end{array} $$

$$\frac{0.32}{1.28} \times 4960^{\prime \prime}=1240^{\prime \prime}=20^{\prime} 40^{\prime \prime}.$$

Wherefore, the corrected value of $E$ becomes $324^{\circ} 37^{\prime} 20^{\prime \prime}-20^{\prime} 40^{\prime \prime}=324^{\circ} 16^{\prime} 40^{\prime \prime},$  with which we repeat the calculation, making use of larger tables.

$$ \begin{array}{ll} \log \sin \varepsilon \quad.\quad.\quad.\quad. &9.7663058n\\ \log e \quad.\quad.\quad.\quad.\quad. &4.7041513\\ \hline &4.4704571n \end{array} \begin{array}{l} \qquad \lambda=29.25\\ \\ \qquad \mu=147 \end{array} $$