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.

36 LUNAR THEORY.
Also by (19), as the Sun's mean motion, since t was = o, is
nt, and therefore his mean longitude = n t + 3 (/3 being his
mean longitude, when t = 0,) and his mean anomaly, consequently, = nt + 3 - Y, we have 0'= Sun's true longitude
= nt + + 2e'. sin (nt + 3 - ).
Now nt+f3=- (0-2esinc0- a) +3:
haà2
or, since the coefficient of 0 must be m,
nt + 3 = mO + t,
(neglecting - 2me. sin ce - a, which is of the second order;). sin (nt + 3 - )) = sin (mO + 3 - -);
0'= (mO +,3) + 2e'. sin (mO + t3 - y),
and (0 - 0') = (i - m)0 3 - 2e'. sin (m0 + - )
2. (0 - 0') = (2 - 2m)O - 2)3 - 4e'. sin (mO + 3 - ),
which we shall call (2 - 2m)O - 23 -p.
49. We have now to find to the first order
sin{(2 - 2m)0-2 - p}, and cos (2 - 2m)0 - 2 3 -p},
p being of the first order.
Now sin {(2 -2m)0 - 23-p}
=sin (2 - 2m) 0 - 23. cosp - cos (2 -2rn) 0 - 23. sinp.
But cosp, or 1 - -+ &c. differs from 1 only by a quanp3
tity of the second order; and sin p, or p - ~ + &c. differs
from p only by a quantity of the third order;.'. sin{(2-2m)0- 2 - p}
=sin (2 - 2m) -2/3 - p. cos (2 - 2m) - 2/3: