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.

FORCES PRODUCING LARGE TERMS IN LONGITUDE. 81
and that by a quantity proportional to the time of action
of these forces. During another equally long time, the
periodic time may be diminished. During the former of
these portions the planet will be dropping behind its mean
place in longitude by an angle which is proportional to the
product of the change in periodic time by the number of
revolutions through which that change is of the same kind:
and as each of these is proportional to the time expressed
by --,, tle planet will have dropped behind its mean
pn - qnI
place by a quantity proportional to (p -qn,) During
the latter portion it will regain the same quantity. The
whole inequality therefore, or the, difference between its actual
place and the place computed on the supposition of mean
motion in longitude, will be proportional to -(,(pn -q n2
On this and sinlilar points the reader is referred to the
author's treatise entitled Gravitation.
PERTURBATIONS OF THE ELLIPTIC ELEMENTS.
102. PRor. 38. To explain algebraically the variation of
parameters.
This artifice of solution is founded entirely upon the
obvious fact, that any equation between two (or more)
quantities (as x and y, or r and 0), may be changed into
any other equation, provided that instead of the constant
quantities we put functions of x and y properly chosen.
For instance, we may change the equation ax + by = c into
the equation (x - e) + (y -f)' = c, provided that instead of
( - e)2 (y -f)2
a we put ( —, and instead of b, (-: or provided
x y
2. (x- e)2 + (y.f)
that instead of a we put ( ), and instead
2x
of b, (' f or in an infinite number of ways. The
2y
constants a and b being sometimes called parameters, the
6