There are several different definitions of conical coordinates defined by Morse and Feshbach (1953), Byerly (1959), Arfken (1970), and Moon and Spencer (1988). The 
system defined in the Wolfram Language is
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
.
Byerly (1959) uses a 
system which is essentially the same coordinate system
as above, but replacing 
with 
, 
with 
, and 
with 
. Moon and Spencer (1988) use 
instead of 
.
The above equations give
 |
(4)
|
 |
(5)
|
 |
(6)
|
The scale factors are
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
The Laplacian is
 |
(10)
|
The Helmholtz differential equation is separable in conical coordinates.
, 
, 
)." §2.16 in 
."
Table 1.09 in