In February 1939 the Gazette published a “diagram for the study and solution of triangles”, in which there was a one-to-one correspondence between each point within a certain area and a triangle of specified shape. The diagram which it is the purpose of this article to explain is even simpler in theory and is perhaps more effective for the solution of triangles, though it has not the interesting features of the one-to-one correspondence between a point and a triangle.

It seems desirable that the methods taught to beginners of reducing the general equation of the second degree to the standard forms should be based on elementary properties of conics directly deduced from the standard equations. The methods given below are suggested as being simpler than those indicated in textbooks or in the references quoted below. It is presumed that pupils are already conversant with the forms and names of the conics represented by the equations (x/p)2±(y/q)2= 1, y2 = 4nx, of the existence of the centre and axes of symmetry (principal axes) of the central conics, and of the axis and the tangent at the vertex of the parabola. No other property of the conic is assumed.

Let the curve be given by x=f(t), y = ɸ(t)

Asymptotes parallel to the coodinate axes.

If as t→tl, y = ɸ(t)→∞ while x=f(t)→a, where a is finite, then x = a is an asymptote.

If as t→t2, x=f(t)→∞while y = ɸ(t)→b, where b is finite, then y = b is an asymptote.

If four rigid rods are smoothly pivoted together at their ends to form a closed quadrilateral the system is said to form a four-link chain. Under certain conditions, five in number, it is found that the quadrilateral is freely deformable and the system is then regarded as a mechanism. Isolated systems, satisfying the necessary conditions and hence admitting deformation, have been known for some time, namely,

• (a) the case in which the rods are coplanar and all the hinges are perpendicular to this plane;

• (b) the general “spheric or conic-chain” in which the lines of the four hinges are concurrent;

• (c) the case of the “Skew-isogram” discovered by Dr. G. T. Bennett.

This article offers suggestions, not for formal lessons on astronomy, but for occasional “raids” into astronomical territory—excursions intended to show how even an elementary knowledge of mathematics may be applied fruitfully to problems of real importance and abiding interest. The mathematical knowledge assumed is acquaintance with the three main trigonometrical ratios and their reciprocals, familiarity with the “natural” trigonometrical tables, and a little practice in changing the subject of a straightforward formula containing the ratios. Values taken to three places generally suffice, and skill in using logarithms, though desirable, is not demanded.

Modern engineering employs the use of the cathode-ray oscillograph as a means to solve some of its problems. A Lissajou figure produced on the cathode-ray tube provides a convenient method of determining the phase difference between two sinusoidal variations of the same frequency. The figure is, in general, an ellipse and this note gives the mathematical basis of the method of determining the phase angle in terms of the axes of the ellipse and of the sides of the principal circumscribing rectangle (whose sides are parallel to the injection axes).

While the conception of the Circle at Infinity as a weapon for analytical investigation has been fairly often applied (as for example “Some equations connected with the Plane Section of the Conicoid” by the author, Math. Gaz., July 1933), its use as the basis for descriptive enquiry has been rarer and disconnected. The aim of this paper is to obtain by simple steps the principal descriptive properties of the quadric surfaces from their intersections with the Plane and Circle at Infinity. The method also has the advantage of making manifest how the basic causes of the differences between the properties of different types of quadric arise, which are often hidden in algebraic treatment.

The object of the present article is to indicate the scope of graphical methods and the desirability of using them far more extensively in teaching elementary mathematics. Articles indicating some of the possibilities have appeared recently in the Mathematical Gazette, one by Dr. W G. Bickley, “Arithmetic, Algebra, and Calculus on the Drawing Board”, May 1941, Vol. XXV, No. 264, and another by C. Dudley Langford, “A graphical method of solving problems on ‘Rate of Work’ and similar problems”, Dec. 1941, Vol. XXV, No. 267. Graphical work is so universal in its application that it is impossible to deal with more than a very small section of work in the compass of a single article, and “Quadratic Equations” has been chosen for purposes of illustration. Through the medium of graphical methods and concrete numerical illustrations, the elementary student of mathematics gets a much clearer and more interesting view of the subject.

The following discussion of the curves r1r2 = crn, where r1, r2, r are the distances of a point from the “foci” ±α, 0 and the origin, c a positive constant and n rational (integral or fractional, positive or negative), provides a good example of extensive knowledge of general curves obtained by elementary-considerations. The attention of the author was first drawn to these curves by the remark of a colleague, Mr. N Davy, M.Sc., that the curves r1r2=cr⅓ appeared in the design of the pole-pieces of certain electromagnets.

A Meeting was held on 4th August. The principal business was connected with the main-school mathematical syllabus, and in particular the new alternative scheme of the Cambridge Local Syndicate. The following points were made:

A fixed sequence is to be deprecated even if it facilitates the task of the examiner; but it was urged against this that the selection of theorems in the Cambridge syllabus, and their arrangement, actually made things easier for the pupil and only restricted the teacher in the area group. It was important to decide whether a formal exposition of geometrical arguments was desirable or not. If it was, it could be practised and tested by a small number of key-theorems such as those suggested by the Syndicate.

A Well-Known method of integrating the simple differential equation of the second order

y″ + k2y = 0 .....1.1

consists in multiplying by an integrating factor 2y′ and integrating directly, when we obtain

y′2 + k2y2 = k2a2, .....1.2

where a is an arbitrary constant.