cke, and both vary from the preceding values.[41]

In the analytical investigation of planetary disturbances, the
disturbing force is usually divided into a radial and tangential force;
the first changing the law of gravitation, to which law the elliptic
form of the orbit is due. The radial disturbing force, therefore, being
directed to or from the centre, can have no influence over the first law
of Kepler, which teaches that the radius vector of each planet having
the sun as the centre, describes equal areas in equal times. If the
radial disturbing force be exterior to the disturbed body, it will
diminish the central force, and cause a progressive motion in the
aphelion point of the orbit. In the case of the moon this motion is very
rapid, the apogee making an entire revolution in 3232 days. Does this,
however, correspond with the law of gravitation? Sir Isaac Newton, in
calculating the effect of the sun's disturbing force on the motion of
the moon's apogee, candidly concludes thus: "Idoque apsis summa singulis
revolutionibus progrediendo conficit 1d 31' 28". Apsis lunae est duplo
velocior circiter." As there was a necessity for reconciling this
stubborn fact with the theory, his followers have made up the deficiency
by resorting to the tangential force, or, as Clairant proposed, by
continuing the approximations to terms of a higher order, or to the
square of the disturbing force.

Now, in a circular orbit, this tangential force will alternately
increase and diminish the velocity of the disturbed body, without
producing any permanent derangement, the same result would obtain in an
elliptical orbit, if the position of the major axis were stationary. In
the case of the moon, the apogee is caused to advance by the disturbing
power of the radial force, and, consequently, an exact compensation is
not effected: there remains a small excess of velocity which geometers
have considered equivalent to a doubling of the radial force, and have
thus obviated the difficulty. To those not imbued with the profound
penetration of the modern analyst, there must ever appear a little
inconsistency in this result. The major axis of a planet's orbit depends
solely on the velocity of the planet at a given distance from the sun,
and the tangential portion of the disturbance due to the sun, and
impressed upon the moon, must necessarily increase and diminish
alternately the velocity of the moon, and interfere with the equable
description of the