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/67

 $$\frac{3 \omega e V(1-e e) \cdot \sin E}{\lambda(1-e \cos E)^{2}} \cdot 206265^{\prime \prime}$$

from which is evident of itself that the error is always circumscribed within uarrow limits when $E$ acquires considerable value, or when $\cos E$  recedes further from unity, however great the eccentricity may be. This will appear still more distinctly from the following table, in which we have computed the greatest numerical value of that formula for certain given values of $E,$ for seven decimal places.

$$E=\begin{array}{r|r} 10^{\circ} & \text { maximum error }= \\ 20 & 3^{\prime \prime} .04 \\ 30 & 0.76 \\ 40 & 0.34 \\ 50 & 0.19 \\ 60 & 0.12 \\ & 0.08 \end{array}$$

The same thing takes place in the hyperbola, as is immediately apparent, if the expression obtained in article 32, VII., is put into this form,

$$\frac{\omega \cos F(\cos F+3 e \sin F) V(e e-1)}{\lambda\left(e-\cos F^{\prime}\right)^{2}} 206265^{\prime \prime}$$

The following table exhibits the greatest values of this expression for certain given values of $F.$

When, therefore, $E$ or $F$  exceeds $40^{\circ}$  or $50^{\circ}$  (which nevertheless does not easily occur in orbits differing but little from the parabola, because heavenly bodies moving in such orbits at such great distances from the sun are for the most part withdrawn from our sight) there will be no reason for forsaking the general method. For the rest, in such a case even the series which we treated in article