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.

34 LUNAR THEORY.
contain cos(2 -2m+c)O-2,-a, and cos (2-2m-c)0-2/3+ta.
But we have seen in (4), that upon solving the equation
dsu
o = + + &c. + A cos (bO + B),
d 02
there will be in the expression for u, a term
-b2 - cos(bO + B).
If then b differs little from 1, there will be a large term in the
value of u. Now 2 -2 m -c is in this case; for c very nearly
=1;.. 2-2m-c=1 - 2m, nearly;.*. (2 -2m -c)2-1
= - 4m, nearly. And, since this term in the differential
equation is of the third order, it will rise in the value of u
to the second order. Our integration therefore to the second
order is not correct, and we must repeat it, examining all
the terms of the third order, and not rejecting those in
which the coefficient of 0 is nearly = 1.
P
46. We may also remark, that the first term of h
which results from the disturbing force, -. will
contain cos (0'- ) or cos (mO + / - Y), nearly, multiplied
by a quantity of the third order. Since m is not nearly
= 1, the resulting term in the expression for u will also
be of the third order. But when, after determining u, we
proceed to integrate the expression
dt 1
dO hu2 I+2 + h
dO I u' (1 +2 ih u')
upon expanding this fraction, there will be one term
C.cos(m0 +f3 - ), C being a quantity of the third order,
the integral of which will give, in the expression for t, a
term C sin (m0 + B - y), the coefficient of which is of the
m
second order. Now the principal object, in the lunar theory,
is to find 0 in terms of t; for which purpose, t must be
found in terms of 0. It will be proper, then, to include