Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

VARIATION OF PARAMETERS. 85
stant, and which was at the extremity of its vibration at
B')
the time -, giving B' the value corresponding to the same
instant. Now we see a distinct and tangible meaning in
the expression. If at any instant t the disturbing force
should cease to act, the penduluni would go on vibrating
so that its arc of vibration on each side would be the value
which A' had at the instant t, and as if it had been at the
B'
extremity of a vibration at the time - where B' preserves
n
constant the value which it had at the instant t. These
two elements (the length of the arc of vibration, and the
time at which the pendulum was at the extremity of the
arc) may, if no disturbing force acts, be found from a
knowledge of the place and the velocity of the pendulum
at any instant: and therefore the variable elements in the
disturbed motion (which, for the instant, are the same) may
be inferred from the place and velocity of the disturbed
pendulum, and may be considered as contained in them.
105. In exactly the same manner we may represent
the motion of a disturbed planet by finding the values of
the elliptic elements, which would represent the place of
the planet at any time t, and which would also give the
velocity and direction of motion of the planet at that time;
the velocity being calculated from those elements as if thev
were the invariable elements of an undisturbed body. And
if the disturbing force should at any instant cease to act,
the planet would go on describing the ellipse whose elements are the values of the elements which correspond to
that particular instant. And the elliptic instafitaneous elements at any time may be inferred from the place and
velocity and direction of motion of the planet at that time,
and may be conceived as contained in them.
106. This method may be used for finding any of the
inequalities produced by perturbation. But it is best adapted
to those which result froin changes in the elements whose
period is very long. For though these changes may be
considerable, so that in a century the orbit of a planet is