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where A is an arbitrary function. This expression can be used directly in (5.73a) to
give

and we impose the condition that be periodic in T. This can be done quite gener-
ally by writing solving for and then imposing periodicity.
However, by virtue of the integral term in (5.75), this produces a somewhat involved
and far-from-transparent result. In order to make some headway, and to produce useful
solutions, let us suppose that the satellite orbit is initially almost circular—a fairly com-
mon situation—so that the value of is small. In particular, the initial conditions give

and so now we are assuming that v is close to 1; it will soon become clear that this
approximation holds as because we will show that decreases to zero from
its initial value.
For small e, we find that

and so we expand equation (5.75) as

where  is a new arbitrary function (replacing A). Now the solution, of equation
(5.76) is periodic in T if

which have solutions (correct to this order in e):

where (which is small).