[14] In this section we will show that it is possible to derive the fields in a sourceless region in the general decomposable medium (16) from two scalar potentials. These scalar potentials are solutions of second-order differential equations. These potentials are generalized Hertz potentials. Let us assume that the two decomposed parts _{c}(**r**) can be derived from two potentials ϕ_{c}(**r**), *c* = *a* or *b*

where _{c}(∇) are six-vector operators that we have to determine. Outside the sources the two potentials satisfy the equations

where we also still have to determine the scalar operators *l*_{c}(∇). To do this we start from the knowledge that outside the sources the fields _{c}(**r**) satisfy (cf. (3) for the equivalent media)

Inserting (57) yields

Comparing this with (58) shows that

with _{c}(∇) six-vector operators that are for the moment still arbitrary. Solving (61) for _{c}(∇) and using (44) and (46) yields

This urges us to chose *l*_{c}(∇) = det_{c}(∇) such that

Note that *l*_{c}(∇) are indeed second-order differential operators. Now we still have to impose that · _{c}(**r**) = 0 or

These are conditions for the six-vector operator _{c}(∇) that can be written compactly as

with

Since _{c,d}^{(2)}(∇) are asymmetric a possible solution of (65) is

where *s*_{c}(∇) are arbitrary scalar operators. This results in

and

The easiest choice for this scalar operators *s*_{c}(∇) are constants *s*_{c}.

[15] Let us now look at the source problem. Suppose that we know that _{c}(**r**) are sources for _{c}(**r**), i.e.,

or using (57)

Using (61) we can recast this further as

Let us now multiply this by arbitrary six-vector operators _{c}(∇). That leads to

If, for simplicity, we take _{c}(∇) and *s*_{c}(∇) constants then we find

a general solution of which can be written as