) that steps have recently been taken at the Observatory,
for calculating the various circumstances of those phenomena, upon the
basis of the more correct data furnished by the researches of Mr.
Adams.--_Translator_.

[34] [Illustration]

The origin of this famous inequality may be best understood by reference
to the mode in which the disturbing forces operate. Let P Q R, P' Q' R'
represent the orbits of Jupiter and Saturn, and let us suppose, for the
sake of illustration, that they are both situate in the same plane. Let
the planets be in conjunction at P, P', and let them both be revolving
around the sun S, in the direction represented by the arrows. Assuming
that the mean motion of Jupiter is to that of Saturn exactly in the
proportion of five to two, it follows that when Jupiter has completed
one revolution, Saturn will have advanced through two fifths of a
revolution. Similarly, when Jupiter has completed a revolution and a
half, Saturn will have effected three fifths of a revolution. Hence when
Jupiter arrives at T, Saturn will be a little in advance of T'. Let us
suppose that the two planets come again into conjunction at Q, Q'. It is
plain that while Jupiter has completed one revolution, and, advanced
through the angle P S Q (measured in the direction of the arrow), Saturn
has simply described around S the angle P' S' Q'. Hence the _excess_ of
the angle described around S, by Jupiter, over the angle similarly
described by Saturn, will amount to one complete revolution, or, 360 deg..
But since the mean motions of the two planets are in the proportion of
five to two, the angles described by them around S in any given time
will be in the same proportion, and therefore the _excess_ of the angle
described by Jupiter over that described by Saturn will be to the angle
described by Saturn in the proportion of three to two. But we have just
found that the excess of these two angles in the present case amounts to
360 deg., and the angle described by Saturn is represented by P' S' Q';
consequently 360 deg. is to the angle P' S' Q' in the proportion of three to
two, in other words P' S' Q' is equal to two thirds of the circumference
or 240 deg.. In the same way it may be shown that the two planets will come
into conjunction again at R, when Saturn has described another arc of
240 deg.. Finally, when Saturn has advanced through a third arc of 240 deg., the
two planets will come into conjunction at P, P', the points whence th