in which
are functions of
alone is called a Stäckel determinant. A coordinate system is separable if it
obeys the Robertson condition, namely that
the scale factors in the Laplacian

(2)

can be rewritten in terms of functions defined by

(3)

such that
can be written

(4)

When this is true, the separated equations are of the form

(5)

The s
obey the minor equations

(6)

(7)

(8)

which are equivalent to

(9)

(10)

(11)

(Morse and Feshbach 1953, p. 509). This gives a total of four equations in nine unknowns. Morse and Feshbach (1953, pp. 655-666)
give not only the Stäckel determinants for common coordinate systems, but also
the elements of the determinant (although it is not clear how these are derived).