1. The applications of spherical harmonic functions in mathematical physics are closely connected with the various potentials—gravitational, electrostatic, magnetic, or the velocity potential in hydrodynamics. It is of interest to recall, briefly, the development of the potential theory. The actual name “potential function” or potential was first introduced by Green in the memoir of 1828 in which he published his famous theorem on the transformation of surface and volume integrals. But the use of the potential function goes back to Euler (1707–1783), who formulated the fundamental differential equations of hydrodynamics, and integrated them in the case where a velocity potential exists, which he recognized to be a special case.

Imagine six drawing pins stuck in the wall. If we stretch threads, either red or blue, between every pair of pins, we inevitably create at least one triangle all of one colour. This we shall call Theorem 1: the proof follows.

Consider any pin. From this emanate five threads (since there are five other pins), of which three must be of one colour. We may without loss of generality assume that they are three red threads.

The usual method of developing the theory of the exponential and logarithmic functions is to take as a definition either

or

The first definition has the disadvantage that with it a good deal of the theory of series is needed. Neither development follows the lines on which the student first met the functions. A good deal could be said for the view that for most students the best type of accurate theory is one which fills up the gaps in methods which are already familiar The difficulties of taking

as a definition are well known.

In ordinary arithmetic a number ab … jk represents the sum of the place values of its digits a, b, …, j, k, these place values being k, 10j, 102i, etc. Also the representation is unique if the digits are restricted to being positive and less than 10. We are going to consider the representation of whole numbers by means of digits with place values, but with a different definition of place value. The digits a, b, …, j, k + 1 of a given number A are to be positive, and the number of them, n, is to be assigned, the units digit k + 1 (in the standard form) is to be greater than zero, and the place value of the digit a in the nth place counting from k + 1 is to be a. a + 1 … a + n - 1 divided by 1.2. …. n, or the number of homogeneous products of a letters taken n at a time; this number or place value is written (a)n.

Since April 2015 there has been no legal requirement in the UK to purchase an annuity with pension savings [1] while for those who reach state pension age on or after 6th April 2016 the UK Government changed the state pension deferral arrangements [2]. The latter refers to an arrangement whereby a pensioner can receive an enhanced state pension by deferring its uptake for an arbitrary number of years. These two changes raise certain questions for prospective pensioners which are worthy of some mathematical consideration.

An annuity is a guaranteed income for life in exchange for a certain sum of money: the pension pot. An alternative to the annuity since April 2015 is a ‘draw down scheme’ in which the pension pot can be used almost like an ordinary bank account and money periodically withdrawn. These two choices arise from ‘defined contribution’ pension arrangements. By contrast ‘defined benefit’ work-based (company) pensions allow no such choice and are not considered further here apart from the special case of the UK state pension. With an annuity a further option to consider, and one which predates the 2015 changes, is whether to take payments that are fixed or index-linked to inflation. Only the UK state pension offers a late retirement enhanced pension if its uptake is deferred.