Axially symmetric mass distributions
At this point, it is convenient to adopt standard spherical coordinates,
, aligned along the axis. These coordinates are related to
regular Cartesian coordinates as follows (see Section A.8):
Consider an axially symmetric mass distribution; that is, a
that is independent of the azimuthal angle, . We would expect
such a mass distribution to generated an axially symmetric gravitational
potential,
. Hence, without loss of generality, we can
set when evaluating
from Equation (3.10). In fact,
given that
in spherical coordinates, this equation yields

(3.27) 
with the righthand side evaluated at . However, because
is independent of , Equation (3.27)
can also be written

(3.28) 
where
denotes an average over the azimuthal angle.
Now,

(3.29) 
and

(3.30) 
where (at )

(3.31) 
Hence,

(3.32) 
Suppose that . In this case, we can expand
as a convergent power series in , to give

(3.33) 
Let us now average this expression over the azimuthal angle, . Because
,
, and
, it is easily seen that
Hence,
Now, the wellknown Legendre polynomials, , are defined (Abramowitz and Stegun 1965b) as

(3.37) 
for
.
It follows that
and so on.
The Legendre polynomials are mutually
orthogonal:

(3.42) 
(Abramowitz and Stegun 1965b).
Here,
is 1 if , and 0 otherwise. The Legendre polynomials also form a complete set; any function
of that is well behaved in the interval
can be represented as a weighted sum of the . Likewise,
any function of that is well behaved in the interval
can
be represented as a weighted
sum of the
.
A comparison of Equation (3.36) and Equations (3.38)–(3.40) makes it reasonably clear that, when , the complete expansion
of
is

(3.43) 
Similarly, when , we can expand in powers of to obtain

(3.44) 
It follows from Equations (3.28), (3.43), and (3.44)
that

(3.45) 
where
Given that the
form a complete set, we can always
write

(3.47) 
This expression can be inverted, with the aid of Equation (3.42), to
give

(3.48) 
Hence, Equation (3.46) reduces to

(3.49) 
Thus, we now have a general expression for the gravitational potential,
,
generated by an axially symmetric mass distribution,
.