Since the end of the last century railway engineers have known that when a railway vehicle passes round a curve a very large force can exist between the outer leading flange and the rail, even under equilibrium conditions. However it was not until Mr. S. R. M. Porter’s paper appeared in 1934 that a satisfactory method of calculating this force was established. Mr. Porter included in his paper two problems of considerable complexity which would take many hours or even days to solve. This may be one of the reasons why his paper is not more widely known.

This product of Queen Anne’s reign consists of five parts: I. Arithmetick, Vulgar and Decimal. II. Algebra or Arithmetick in Species. III. The Elements of Geometry. IV. Conick Sections. V. The Arithmetick of Infinites explained, with its application to Superficial and Solid Geometry. To these is added an Appendix of Practical Gauging, and in the eighth edition is also found the History of Logarithms.

Here is a published example of what I mean by the sinusoidal fallacy:

Interesting algebraic and geometric results can be obtained by interleaving integer sequences term by term. To introduce this we consider how a well-known sequence can be considered as being composed of subsequences.

The standard Fibonacci sequence (Fk):

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... (1)

is defined by the recurrence relation Fk = Fk - 1 + Fk - 2 for k ≥ 3 and F1 = F2 = 1.

The work presented here was prompted by curiosity concerning a structural detail of the ironwork supporting the overhead cover of the railway platform at Stockport, Cheshire. The detail – see the front cover which shows the roof support – comprises an elliptical ‘quadrant’ joined to vertical and horizontal members of the supporting structure, with a circular ring fitted interstitially so as to touch the elliptical arc and the two straight members. The question arises: what is the radius of the circular ring in terms of the parameters of the ellipse?

1. Introduction

In the article [1], a follow-up to my article [2], Christian and Trustrum cite empirical evidence that the probability of a family giving birth to a boy or to a girl may vary from family to family. Letting bi and gi = 1 − bi respectively denote the probability of the ith family giving birth to a boy or to a girl, the suggestion is that bi may be distinct from bj for distinct i and j. As they point out, this has implications for the expected society-wide number of boys and girls. Following my discussion of the ‘stop after k girls’ policy, Christian and Trustrum introduce the related policy ‘stop after k girls or N children’. They argue that the expected number of children under this latter policy is an increasing function of bi (when 0 < k < N). The present article complements their discussion by examining this alternative stopping policy in more detail.

Integration is an important topic discussed in first year calculus. Many subsequent topics, such as solving first-order or second-order differential equations, requires the evaluation of at least one integral, and sometimes several.

The most challenging material in integration is arguably the techniques of integration. Unfortunately, due to time constraints, only basic techniques are commonly taught, while in reality, many integrals having special forms are solvable only by using more sophisticated techniques.