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.

26 LUNAR THEORY.
36. It appears, then, that upon substituting these
values in the equation (d), it will be reduced to this form,
d'u
- + u + n = 0, nI being a complicated function of u, s,
du2
and 0. No method of directly solving such an equation
is known: but we have seen in Prop. 1, that it could be
solved, if Il were a function of 0 only. This suggests the
method of solving by successive substitution. Find a value
of u in terms of 0, which is nearly the true one: substitute this value for u, in the terms of small magnitude;
II will then be a function of 0 only, and the equation
may be solved, and a more approximate value of u found.
Substitute this for u in I, and again solve the equation,
and a value will he found still nearer the truth. Proceed
in the sane manner to find the value of s.
37. But, in order to carry on this process with facility,
it is necessary to establish some rule with regard to the
comparative value of small quantities, so that, fixing upon
some quantity as a standard, our first approximation may
include its first power, and the first powers of quantities
nearly as great; our second approximation may comprehend
its square, and the squares of the others, and the products
of any two, &c. Thus, let e be the eccentricity of the
lunar orbit: e' that of the solar orbit; k the tangent of
the mean inclination of the lunar orbit to the ecliptic: m
the ratio of the Sun's mean motion to the Moon's mean
1! 1 1 1
motion*. Here e =- nearly; e = -; k=-; m=-:
20 12 13
taking e, then, as our standard, e', k, and m, are small
quantities, not differing much in magnitude from e, and are
therefore said to be small quantities of the first order. But,
P Z'. i
or - is little more than -, and therefore admits better
r' zu 400
of being compared with e2 than with e: it is on that account
considered to be a small quantity of the second order:
mne, P e, &c. would be called of the third order; &c.
r
*By the term mean motion is meant, the velocity with which the mean longitude increases. The mean motion varies therefore inversely as the periodic time.