In this formalism, is called the generating function and is called the pilot vector.
The choice of generating function is determined by the symmetry of the scalar equation,
i.e., it is chosen to solve the desired scalar differential equation. If is taken as

(25)

where
is the radius vector, then is a solution to the vector wave equation in spherical coordinates.
If we want vector solutions which are tangential to the radius vector,

(26)

(27)

(28)

so

(29)

and we may take

(30)

(Arfken 1985, pp. 707-711; Bohren and Huffman 1983, p. 88).

A number of conventions are in use. Hill (1954) defines

(31)

(32)

(33)

Morse and Feshbach (1953) define vector harmonics called , , and using rather complicated expressions.