Page:DeSitterGravitation.djvu/18

The periodic terms, all of which have very short periods, can be entirely neglected. For the law I. we find that there are no secular terms not multiplied by $\mu '$, which is in accordance with what was found above. The terms which have $\mu '$ as a factor also give nothing secular in $\delta a$ or $\delta e$. In $\delta \varpi$ we have—

$\delta \varpi =2\mu '{\frac {\lambda ^{2}}{p}}v.$

For the law II. we find this same term, but also a secular term not multiplied by $\mu$, viz.—

$\delta \varpi ={\frac {1}{2}}{\frac {\lambda ^{2}}{p}}v.$

(36)

With the law II. we have thus a secular motion of the perihelia, even for a planet of infinitesimal mass.^[1] The motion in one century is—

For	Mercury[2]	: $\delta \varpi$ = +	7″.15	$e\delta \varpi$ = +	1″.470	(37)
"	Venus	:	1.43		0.009
"	the Earth	:	0.639		0.011
"	Mars	:	0.225		0.021
"	Jupiter	:	0.0104
"	Saturn	:	0.0023
"	Uranus	:	0.00040
"	Neptune	:	0.00013

For	☄ Encke	: $\delta \varpi$ = +	0″.308
"	☄ Halley	:	0.0071

The effect of the secular terms in $\delta \epsilon _{1}$ is, of course, only to make the observed mean motion disagree with the value derived from Kepler's third law. As we have no sufficiently accurate means of independently determining the mean distances, this can never be detected.

The formulæ (35) are valid for all excentricities, and can thus also be used for comets. The perturbations in $\delta \varpi$ remain of the same order as for planets. Those in $\delta e$ and $\delta \epsilon _{1}$ become zero for a parabola. Only the perturbation in $\delta a$ would appear to become very large at perihelion. We have, however,

$\delta p=\left(1-e^{2}\right)\delta a-2ae\delta e,$

from which we find, for both laws,

$\delta p=-2\lambda ^{2}\left(1-2\mu '\right)e\ \cos \ v,$

which does remain small for $e=1$.

1. This was also found by Mr. Wacker, who gives for Mercury 7".2 per century (l.c., page 58).
2. The value 6″.69 quoted by Professor Lorentz in his Göttingen lecture (l.c., page 1239) was a first approximation.