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/71

 [1], article 37 ; the second, the determination of $E$ from $A$  and $B,$  which may be done directly, either by means of the equation

or by this,

$$E=2 B\left(A^{\frac{1}{2}}+\frac{1}{15} A^{\frac{3}{2}}\right)$$

$$\sin E=2 B\left(A^{\frac{1}{2}}-\frac{3}{5} A^{\frac{3}{2}}\right)$$

the third, the determination of $v$ from $E$  by means of equation VII., article 8. The first operation, we will bring to an easy calculation free from vague trials; the second and third, we will really abridge into one, by inserting a new quantity $C$ in our table by which means we shall have no need of $E,$  and at the same time we shall obtain an elegant and convenient formula for the radius vector. Each of these subjects wè will follow out in its proper order.

First, we will change the form of equation [1] so that the Barkerian table may be used in the solution of it: For this purpose we will put

from which comes

$$A^{\frac{1}{2}}=\tan \frac{1}{3} w \sqrt{\frac{5-5 e}{1+9 e}}$$

denoting by $\alpha$ the constant

$$75 \tan \frac{1}{8} v w+25 \tan \frac{1}{2} w^{3}=\frac{75 k t \sqrt{ }\left(\frac{1}{5}+\frac{1}{b} e\right)}{2 B q^{\frac{3}{2}}}=\frac{\alpha t}{B},$$

$$\frac{75 k \sqrt{ }\left(\frac{1}{5}+\frac{9}{5} e\right)}{2 q^{\frac{3}{2}}} .$$

If therefore $B$ should be known, $w$  could be immediately taken from the Barkerian table containing the true anomaly to which answers the mean motion $\frac{\alpha t}{B} ; A$  will be deduced from $w$  by ineans of the formula

$$A=\beta \tan ^{2} \frac{1}{2} w$$

denoting the constant

$$\frac{5-5 e}{1+9 e} \text { by } \beta \text {. }$$

Now, although $B$ may be finally known from $A$  by means of our auxiliary table, nevertheless it can be foreseen, owing to its differing so little from unity, that if the divisor $B$  were wholly neglected from the beginning, $w$  and $A$  would be affected with a slight error only. Therefore, we will first determine roughly $w$ and $A,$  putting $B=1;$  with the approximate value of $A,$  we will find $B$  in our