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.

EXPANSION OP TERMS IN THE EQUATION. 39
51. PROP. 18. To find the values of
'T T d2u \ T T d u
J2IW ~' kW d Of + u J and
h2?43 ' d2 ) 24 h2U3 02s k and d.
to the third order.
T
The only term of -2 to be taken here, is
3 m'u"3
--— 2. sin 2. (0 - 0'),
(the others being of the fourth order), or, as in (50),
3
- m21 + e'cosmO + 3 - 4ecoscO - a} x
{sin (2 -2m) - 23 - 2e'. sin (2 - m) - 3 -
+ 2e'. sin (2 - 3m) 0 - 3$3 + },
which
3e'
sin (2 - 2m) 0-2 /3+ - sin (2 - ) 0 - -
e'
+ - sin (2 - 3 m) 0 - /3 +
_ - 3 2 - 2e sin (2-2m + c) 0-2/3 - a
- 2e sin (2 - 2 - c) 0 - 2/3 + a
- 2e'sin (2 -m)0-/3-
+ 2e' sin (2 - 3 mn)O-33+.
52. Now, since u = a (1 + e coscO - a),
du
d = - cae sin (cO - a) =- ae sin (cO - a), very nearly;
du
As this is a quantity of the first order, we shall use only
T
the first term of - to multiply with it (because the proh2u3
duct of ae by the other terms would be of the fourth
order);