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EELATIONS BETWEEN THE ECCENTEICITIES AND INCLI- 
NATIONS OF THE OEBITS OF THE PLANETS JUPITER 
AND SATURN. 
By R. T. a. Innes, F.R.S.S.Af. 
(Read June 21, 1911.) 
It is well known that the eight major planets revolve around the Sun 
in nearly the same plane and in elliptical orbits which do not depart 
greatly from circularity. 
In the cases of the Earth, Mars, Jupiter, and Saturn, the equators of 
these planets are not greatly inclined to the common plane of the orbits ; 
the equators of Mercury, Venus, Uranus, and Neptune have not been 
seen, but from the motion of the satellites of Uranus and Neptune it is 
inferred that their equators have large inclinations. 
The precise calculations of astronomers, which unfortunately can only 
extend to a few hundreds of years, agree with observation in showing that 
the eccentricities and inclinations of the various orbits are changing 
slowly. Lagrange, Laplace, and Poisson have proved that the mean- 
distances of the planets are essentially invariable, and the two former 
have, by very rough and inexact methods, shown that the sums of simple 
functions of the eccentricities and inclinations of all the planets will 
always remain small, but on account of the preponderating masses of the 
four outer planets the equations prove nothing as to the stability of the 
orbits of the four inner planets, of which group the Earth is one. It 
cannot to-day be proved that the orbit of such a planet as the Earth is 
stable. The large uniformity of flora and fauna for great ages gives rise to 
a no doubt well-founded belief that the Earth's orbit is stable, although 
there may be no mathematical proof yet available. 
The case is different with the planets Jupiter and Saturn ; the pertur- 
bations which these planets undergo, due to the presence of the other 
planets, are so insignificant compared to their own action on each other 
that they can be considered apart. In fact, the preponderating masses of 
the Sun, Jupiter, and Saturn in the solar system make these three bodies 
a nearly ideal case of the " problem of three bodies," and in consequence 
