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.

FURTHER APPROXIMATION. 55
70. In the same manner in which we have approximated
to the values of t and 0 to the second order, we might go
on to the third and higher orders. For the third order, it
would be necessary to examine the terms of the equation to
the fourth order, and thus the last terms, in the expressions
p
for h, &c., in Art. 35, would be employed. As the method
of conducting all these approximations must be the same, wé
shall here mention the several steps.
(1) From the last approximate value, find t in terms of 0.
(2) Since O' is, by the elliptic theory, found in terms of
t, it can be expressed in terms of 0, and - 0', 2 (0 - 0'), &c.
can be expressed. u' also, which is known in terms of 0', can
be expressed in terms of 0.
(3) Find expressions for sin 2 (0 - 0'), cos 2 (0 - 0'), &c.
to as many orders as may be necessary.
(4) Substitute these values, and the last approximate
P
value of u, in the expressions for h2- '&c.
(5) When these are substituted in the equation, integrate it, as for the second order.
(6) Proceed in the same way in every respect, for the
determination of s.
71. In carrying the approximation to higher orders, it
frequently happens, that a. term will rise, by integration, two
orders*. This renders the operations very troublesome, and
particular methods are sometimes necessary: but we cannot
stop to explain them here.
72. We shall here mention some of the most interesting
results of the next approximation. (1) The last terms in the
expressions for 2 —, &c. introduce into the equation for u
' An instance of this will be seen in the Tract on the Figure of the Earth,
Art. 71.