Binal Fractions. The reciprocals (1/N) of any number (N) will be termed Binal Fractions, or shortly Binals, when expressed in the binary scale. The notation used is similar to that of ordinary decimals. Thus (see table at end):

1°. A point is used to mark the beginning of the fraction.

2°. The binal (1/N) consists of a certain sequence of ciphers and units (the only digits used) to the right of the point; partly non-recurrent and partly recurrent.

The solution of the general case of a cable having constant resistance R, inductance L, capacitance C, and leakance G, all expressed per unit length, is usually obtained in the form of an expansion. Using his operational method, Heaviside was the first person to obtain a solution. F. W Carter points out that Heaviside’s result seems to be incorrect, and proceeds to find a solution in the form of a double summation. By aid of complex integration, the integrand being expanded in Bessel functions, H. Jeffreys gives another form of series solution. As this type of problem may arise in lectures on applied mathematics, it is proposed to show how a compact result can be got by contour integration, using a suitable transformation.

Since due recognition is now being more and more widely given to the importance of Vector Analysis for three-dimensional work in mechanics, geometry and mathematical physics, the time is opportune to consider the place which that subject merits in a University degree course. A student who undertakes anything like research work in applied subjects is considerably handicapped if he has had no training in Vector Analysis; but the need for it is felt much earlier than the stage at which research is generally begun. Every University course contains subjects which are less useful than Vector Analysis; and on the ground of utility alone there is no question that the latter deserves a place in the curriculum. It will, however, be found quite unnecessary to displace anything else, because the time occupied in teaching the essential parts of vector algebra and calculus will be saved, even during a three years’ course, by the application of this method to the three-dimensional parts of mechanics and mathematical physics.

In technical colleges one is often called upon to introduce advanced mathematical techniques to students whose background is not very extensive. Such a method is the use of the Laplace Transform for the solution of linear differential equations with constant coefficients. Textbooks which deal with this topic, even those specifically written for engineers, derive the transform from the Fourier Integral, or from Heaviside’s Operational Calculus, or just brusquely define the process. Then certain standard forms are evaluated, the rules for application to differential equations are laid down and a selection of worked examples follows. But such a treatment can mystify a student who wants to know why, in following these rules, he should, in effect, firstly, multiply each term of the equation by e-px, secondly, integrate the resultant products from 0 to ∞, and, thirdly, obtain in this way the complete solution.

David Pagni drew attention to a result which is ascribed by Dickson [2, p. 286] to Liouville (1857), that the sum of the cubes of the number of divisors of each of the divisors of an integer, is equal to the square of their sum. For example, the divisors of 6 are 1, 2, 3, and 6, which have 1, 2, 2, and 4 divisors respectively, and

Pagni observed, as have others, including Mason et al. [4, p. 179], that the reason Liouville’s result works is because the numbers generated are element-by-element products of sequences of the form {1, 2, ..., t}, which are well known to have the sum of their cubes equal to the square of their sum.

The world high jump record for men is now an amazing 2.45 m (8 ft 0½ in). This requires an athlete to raise his or her the centre of gravity through a vertical distance of about 1.3 metres, something which is quite impossible by leg thrust alone. Through experience, coaches and athletes have developed techniques which maximise performance, and this paper attempts to describe these in mathematical terms. In this article, the various factors contributing to the height of a jump are investigated with the aid of information gained from a film of a Straddle-style jump. The same principles are involved in a jump using the Fosbury Flop.