be readily understood, when we state that the disturbing action of the sun and moon upon the terrestrial spheroid produces an oscillation of the earth’s axis which occasions a periodic variation of the obliquity of the ecliptic.  Now, as the plane of the ecliptic approaches the equator, the mean disturbing action of the sun and moon upon the redundant matter accumulated around the latter will undergo a corresponding variation, and hence will arise an inconceivably slow movement of the plane of the equator, which will necessarily affect the obliquity of the ecliptic.  Laplace found that if it were not for this cause, the obliquity of the ecliptic would oscillate to the extent of 4 deg. 53’ 33” on each side of a mean value, but that when the movements of both planes are taken into account, the extent of oscillation is reduced to 1 deg. 33’ 45”.

Variation of the length of the tropical year.—­The disturbing action of the sun and moon upon the terrestrial spheroid occasions a continual regression of the equinoctial points, and hence arises the distinction between the sidereal and tropical year.  The effect is modified in a small degree by the variation of the plane of the ecliptic, which tends to produce a progression of the equinoxes.  If the movement of the equinoctial points arising from these combined causes was uniform, the length of the tropical year would be manifestly invariable.  Theory, however, indicates that for ages past the rate of regression has been slowly increasing, and, consequently, the length of the tropical year has been gradually diminishing.  The rate of diminution is exceedingly small.  Laplace found that it amounts to somewhat less than half a second in a century.  Consequently, the length of the tropical year is now about ten seconds less than it was in the time of Hipparchus.

Limits of variation of the tropical year modified by the disturbing action of the sun and moon upon the terrestrial spheroid.—­The tropical year will not continue indefinitely to diminish in length.  When it has once attained a certain minimum value, it will then increase until finally having attained an extreme value in the opposite direction, it will again begin to diminish, and thus it will perpetually oscillate between certain fixed limits.  Laplace found that the extent to which the tropical year is liable to vary from this cause, amounts to thirty-eight seconds.  If it were not for the effect produced upon the inclination of the equator to the ecliptic by the mean disturbing action of the sun and moon upon the terrestrial spheroid, the extent of variation would amount to 162 seconds.