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/36

 9.

If a perpendicular let fall from any point whatever of the ellipse upon the line of apsides is extended in the opposite direction until it meets the circle described with the radius $a$ about the centre of the ellipse, then the inclination to the line of apsides of that radius which corresponds to the point of intersection (understood in the same way as above, in the case of the true anomaly), will be equal to the eccentric anomaly, as is inferred without difficulty from equation IX. of the preceding article. Further, it is evident that $r \sin v$ is the distance of any point of the ellipse from the line of apsides, which, since by equation VIII. it $=a \cos \varphi \sin E,$ will be greatest for $E=90^{\circ},$  that is in the centre of the ellipse. This greatest distance, which $=a \cos \varphi=\frac{p}{\cos \varphi}=\sqrt{a p},$ is called the semi-axis minor. In the focus of the ellipse, that is for $v=90^{\circ},$ this distance is evidently $=p,$  or equal the semi-parameter.

10.

The equations of article 8 comprise all that is requisite for the computation of the eccentric and mean anomalies from the true, or of the eccentric and true from the mean. Formula VII. is commonly employed for deriving the eccentric from the true; nevertheless it is for the most part preferable to make use of equation X. for this purpose, especially when the eccentricity is not too great, in which case $E$ can be computed with greater accuracy by means of X. than of VII. Moreover, if X. is employed, the logarithm of sine $E$ required in XII. is had immediately by means of VIII.: if VII. were used, it would be necessary to take it out from the tables; if, therefore, this logarithm is also taken from the tables in the latter method, a proof is at once obtained that the calculation has been correctly made. Tests and proofs of this sort are always to be highly valued, and therefore it will be an object of constant attention with us to provide for them in all the methods delivered in this work, where indeed it can be conveniently done. We annex an example completely calculated as a more perfect illustration.