Abstract
In addition to harmonic functions, which were considered at length in chapter II, it is also of interest to consider harmonic vectors H; i. e., vector fields satisfying the pair of equations (1) such a field is termed harmonic because each of its three components is a harmonic function and the components are connected by generalized anti-Cauchy-Riemann equations. We shall obtain for such fields an integral representation which is a natural generalization of the representation (II. 3.1) of harmonic functions. If we denote a vector field H by means of its cartesian components, H = (H 1, H 2, H 3), we readily see that if one of its components, say H 1, is a given harmonic function, the equations (1) constitute a system of first-order partial differential equations which must be satisfied by the other two components. However, this system of equations does not completely determine the remaining two components. In fact, the following lemma holds:
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