Application of a polar form of complex quantities to the calculation of alternating-current phenomena

In the calculation of alternating current phenomena by means of complex quantities, as a rule, the rectangular components of the vector are used, and the rectangular form involving the operator is more common than the polar or exponential forms which involve the operators (cos θ + j sin θ) or j^θ; although it is recognized that the latter are very convenient in certain cases. A simple method for dealing directly with the vectors themselves is described in the paper and it consists in introducing the operator jⁿ, where n, contrary to ordinary usage, may be any positive or negative fraction. Just as j or j¹ rotates the quantity before which it is placed through 1 × 90 degrees, so jⁿ rotates the number into which it is multiplied through n × 90 degrees. The operator jⁿ follows the rules of ordinary algebra and according to these the different algebraic operations of multiplication etc., are developed in section II. In section III a few illustrative problems are given; these are followed by a critical resume in section IV. At the end, for convenience of reference a summary of formulas is given, and a very short bibliography is included.