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The approach to plane mechanics by vector methods is often considered to be an unnecessarily sophisticated one and it is true that the preliminary work in vector algebra is lengthy and out of keeping with the elegance achieved. The natural tool, it has been said, is the complex number, and I shall obtain some of the fundamental results by using only complex number theory and vector addition.

The object of this article is to show how the problems of transformation geometry, and in particular the determination of the composition of two or more successive transformations, can be solved far more easily by Argand diagram methods leading to algebraic equations involving complex numbers, rather than by purely geometric considerations. It must be conceded, of course, that the methods advocated here cannot be extended to three dimensions.

Contemporary writers of textbooks on electricity and magnetism and on hydrodynamics make considerable use of the complex variable in two-dimensional problems. It seems strange that two-dimensional dynamics is not taught in this way, even though the application of vectors to three-dimensional dynamics has now become respectable. Mr. Buckley's article preceding this has shown how the basic results may be established. The following examples are further indications of how this powerful technique may be profitably employed in two-dimensional kinematics. The methods are essentially labour-saving.

We denote the set of all real numbers by ℝ. It is assumed that each reader is acquainted with the real number system and all its properties. In particular we assume a knowledge of the ordering of ℝ, the definitions and properties of the supremum and infimum (sup and inf), and the completeness of ℝ (every non-empty set in ℝ which is bounded above has a supremum). It is also assumed that every reader is familiar with sequential convergence in ℝ and with infinite series. Finally, no one should undertake a study of Complex Variables unless he has a thorough grounding in functions of one real variable. Although it has been traditional to study functions of several real variables before studying analytic function theory, this is not an essential prerequisite for this book. There will not be any occasion when the deep results of this area are needed.

Recent work on complex number plane mapping as an aid to muitivariable control system design is adapted to deal with discrete systems in particular those of the second order. It is shown how all the known results from continuous theory are immediately available for the discrete case. Formulae are derived for calculating the feedback gain factors for a second-order system once the desired eigenvalue locations have been specified.

Investigations into a technique for mapping the state space onto the complex number plane have recently been carried out by Macfarlane, Sabouni and Munro. This paper presents the collective results of those investigations. The theory of the mapping is developed and the properties of the resulting complex plane representations are examined, in particular an additive property. An extension to existing eigenvalue bound theorems is suggested by the mapping and is exploited in the use of the mapping as a design aid.

In [1] the author offered the following postulational basis for an algebraic structure called a heaviside operational calculus. Let G be an additive Abelian group of operands. Let E be a set of endomorphisms on G. If a nonnull element, U, of G has the property that every element of G is the map of U under some endomorphism in E, we call U a unit element of G relative to E. Suppose G and E satisfy the following postulates: 1. The endomnorphisms in E are permutable. 2. G contains at least one unit element, U, relative to E. 3. Any endomorphisin in E that sends a unit element U into a nonnull operand sends every nonnull operand in G into a nonnull operand. It follows, as shown in [1], that E is an integral domain under the usual definitions of equality, sums, and products for endomorphisms. For some choices of G and E, E will also be a field. G and E (or G and a class of operators associated with E) determine a heaviside operational calculus. For example, let G be the class of entering, sectionally continuous functions of t. The unit step function, {u(t) }, will serve as a unit operand for postulate 2. Given a member, f, of G we find the following endomorphism on G that sends u intof

(1967). On the Automorphisms of the Complex Number Field. Mathematics Magazine: Vol. 40, No. 4, pp. 213-213.

Summary Newton's method for the evaluation of the zeros of analytic functions is generalized by the recurrence relation zi+i = Zi- (ki-li)fi\zi)/fli+1)(zi), where ki and k are determined so that ideally U + 1 = ki = k in the vicinity of a zero of order k. The process so defined is a second-order one and yields accurate approximations to zeros for all values of k, provided I,, + 1 = k{ = k.