The study of the function of a real variable is aided by the " graph " of the function. Whenever difficult analytical notions are considered, such as continuity, turning points, etc., a study of the graph enables the student to get a general view of the problem.

No such help appears to be available in teaching the theory of functions of the complex variable. A " graph " of f(x + iy) = u + iv would require a fourdimensional space. Nevertheless some representation of an analytic function seems necessary if the student is not to be discouraged by the abstract nature of the idea.

1. The discussion of the motion of bodies of varying mass recently revived by Mr. A. S. Ramsey's article in the Gazette for July 1941 has produced many sidelights on the momentum method of treating these problems, but not one of your correspondents has suggested the alternative approach by the method of energy. This method was shown to me (and, if I am not greatly mistaken, also to Mr. Ramsey) by one of the ablest, if least known, of the last generation of Cambridge teachers of mathematics, William Welsh of Jesus, but it does not yet seem to have reached the textbooks. It has one great advantage over the momentum approach in that it forces attention on the physical assumptions involved, the slurring of which is the main cause of the difficulty experienced by many students in applying the equations of momentum.

In the preparation of mathematical diagrams for reproduction, it is often necessary to draw an ellipse or parabola with considerable accuracy. The methods described below need only a ruler, compasses and a set of French curves ; they produce very satisfactory results with a minimum of construction.

The process consists in setting out, by simple geometrical constructions, short circular arcs of curvature and then filling in the spaces between them with the aid of French curves. It is best to do the actual constructions in pencil and to ink the curves in afterwards with a drawing pen and French curves.