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.

84 PLANETARY THEORY.
B'= B- sin (nt+pt - B-Q)
2n (n + p)A
2n(n - p)A
a form which is sometimes more convenient than that obtained by direct solution.
104. PIOP. 39. To explain geometrically the variation of parameters.
This can be done most readily by reference to some
instance: for example, that of the last Article. The
equation
d2Z,
4- n2~ = 0,
dt2
is the equation corresponding to the motion of a simple
pendulum whose length is n2g, in a cycloidal arc. The
equation
t n2 + acos (p t - Q) =
represents the motion of the same pendulum, supposing that
besides the force of gravity there is a disturbing force
a.cos(pt - Q) acting upon it in the direction of a tangent
to the curve. The solution z'= A'. cos(n t - B'), if we look
no further, merely asserts that the place of the pendulum
at the time t may be expressed by means of the same formula as that which expresses the place in undisturbed vibration. Of this there is no doubt, as the same thing might be
expressed in the same form with a hundred different values
of the parameters. But if we differentiate it, we find (by
the assumption of last Article) that the velocity, with the
values of A' and B' that we have found, is -nA'sin(nt-B'):
that is, that the velocity of the body also is expressed by
the same formula as if its vibration were not disturbed by
the additional force. Consequently the place and velocity
of the disturbed pendulum at the time t are the same as
those of an undisturbed pendulum, whose arc of vibration
on each side of the vertical is the value of A' at that in