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/28

 IV. For different bodies moving about the sun, the squares of these quotients are in the compound ratio of the parameters of their orbits, and of the sum of the masses of the sun and the moving bodies.

Denoting, therefore, the parameter of the orbit in which the body moves by $2 p$, the mass of this body by $\mu$ (the mass of the sun being put $=1$ ), the area it describes about the sun in the time $t$  by $\frac{1}{2} g,$  then $\frac{g}{t \sqrt{p} \sqrt{1+\mu}}$  will be a constant for all heavenly bodies. Since then it is of no importance which body we use for determining this number, we will derive it from the motion of the earth, the mean distance of which from the sun we shall adopt for the unit of distance; the mean solar day will always be our unit of time. Denoting, moreover, by $\pi$ the ratio of the circumference of the circle to the diameter, the area of the entire ellipse described by the earth will evidently be $\pi \sqrt{p}$, which must therefore be put $=\frac{1}{2} g,$  if by $t$  is understood the sidereal year; whence, our constant becomes $=\frac{2 \pi}{t \sqrt{1+\mu}}$. In order to ascertain the numerical value of this constant, hereafter to be denoted by $k,$ we will put, according to the latest determination, the sidereal year or $t=365{.}2563835,$  the mass of the earth, or $\mu=\frac{1}{354710}=0{.}0000028192,$   whence results

$$\begin{aligned} \log 2 \pi \quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad 0.7981798684\phantom{.} \\ \operatorname{Compl. log.} t \quad.\quad.\quad.\quad.\quad.\quad 7.4374021852\phantom{.} \\ \operatorname{Compl. log.} \sqrt{1+\mu} \quad.\quad.\quad.\quad 9.9999993878\phantom{.} \\ \log k \;\;\quad.\quad.\quad.\quad.\quad.\quad.\quad.\quad 8.2355814414\phantom{.} \\ k= \qquad \qquad \qquad 0.01720209895. \end{aligned}$$

2.

The laws above stated differ from those discovered by our own {{sc|Kepler in no other respect than this, that they are given in a form applicable to all kinds of conic sections, and that the action of the moving body on the sun, on which depends the factor $\sqrt{1+\mu}$, is taken into account. If we regard these laws as phenomena derived from innumerable and indubitable observations, geometry shows what action ought in consequence to be exerted upon bodies moving about