A set of curvilinear coordinates defined by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
,
,
and 
.
There are several notational conventions, and whereas 
is used in this work, Arfken (1970) prefers 
. The following identities show that curves of constant
and 
are circles in 
-space.
 |
(4)
|
 |
(5)
|
The scale factors are
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
The Laplacian is
 |  |  |
(9)
|
Laplace's equation is not separable in bipolar cylindrical coordinates, but it is in two-dimensional bipolar coordinates.
, 
, 
)." §2.9 in