where A(r), C(r) are analytic functions of the real variable r = x^Xμ. [μ = 1, 2, 3] (Repeated indices mean the summation convention is used.) In this paper we shall investigate the four variable analogue of this equation, TA[W] = 0, and show that many of Bergman's results carry over to this case. Here, we need in many instances, the methods of several complex variables in order to find the natural generalizations. In Bergman's theory, the integral operator B3 [/] plays an important role in studying the solutions of (1.1). In our case, there is an analogous operator [7]~[1O], which is a four-variable analogue to