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the roots z may be real or complex numbers. There are various ways of solving the problem: a. In one group of methods, estimated approximations of z are substituted in Eq. (1) and varied by trial and error until the functional value of this equation becomes zero. If z is complex, the computational labor increases considerably. To alleviate it, several electronic mechanical computers of the analogue type have been developed, e.g. by H. C. Hart and J. Travis [1]. A modification of this group by W. R. Evans [2] applies the special concepts of the theory of servomechanisms. b. The well known root squaring method of Graeffe transforms the coefficients a, of Eq. (1) gradually into a sequence, identical with the higher squared root values. However the serious disadvantage of this method is that it yields only the moduli of the complex roots, whereas the phase angles have to be determined separately by additional methods given by F. W. J. Olver in [3]. c. The process of factorization splits Eq. (1) into separate factors giving immediately the root values. For linear factors, i.e. real roots, the process is identical with the wellknown Regula Falsi, Newton's rule and Horner's method. For quadratic factors, which include conjugate complex roots, the procedure is due to L. Bairstow [4]: starting with an estimated approximation of the root factor, the coefficients a, of Eq. (1) are transformed by long division into the coefficient set &,■ . The remainder terms of bt , together with an additional coefficient set c, derived from the 6, , are used to calculate corrections for the root parameters. The 6,and c,-sets are computed by reduction formulae suitable for routine work. The method represents an extension to Newton's rule for the complex. It was later systematized by W. E. Milne [5]. The procedure by S. N. Lin [6] corresponds analogously to a simplified correction formula; however it is convergent only for restricted cases. The above factorization process is extended here by the following results.

Determination of the root systems of algebraic equations by affinity transforms