The general problem. This paper deals with particular cases of the following problem: A rectangular array of square cells is given; a particle P moves from one cell to another in such a way that, when it is in any cell, it is equally likely to pass across any of its edges. P is liberated in any given cell, and it is required to find the probability that it will ultimately emerge across any stated edge in the boundary of the array.

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.

If P is a point of intersection of two conics which have one common focus, then e1 |PM1| =e2 |PM2|. It follows that if the two directrices cut in O, then OP passes through a second point of intersection, and that if directions of measurement perpendicular to the directrices are assigned, the equations e1 . M1P = ±e2M2P give two of the chords of intersection, we may say that these are the chords associated with the common focus.

Several proofs, based on projective considerations, have been given recently of the Petersen-Morley theorem. In this note a direct metrical proof is given, based on the dual vectors of Study, which seems simpler than any of these.

The design shown in Fig. 1 is unusual in the amount of pattern that

goes to one repeat, only one complete repeat being included in the illustration. It is also unusual in that it includes fifteen-rayed stars.

Nothing can convince us more that “things are not what they seem” than a comparison of the descriptions by two men of what they remember of Cambridge in the seventies ! I should of course recognise my friend Professor Forsyths picture of the Cambridge of that time and am grateful for his calling to my mind so many memories of the past. But for me he has painted them in far too drab a colour, so that one wonders whether his undergraduate days could have been like mine some of the happiest in his life. The reading of his paper enforces upon me the truth of the principle that environment and personalities are not things in themselves, but are our own “constructs”—each of us giving our own “reality” to such evasive phenomena.

1. When we proceed to obtain the equation of the join of points on the ellipse x2/a2 + y2/b2 = 1 whose eccentric angles are θ and ϕ, we find that cos ½ (θ − ϕ) is a factor of each term, and similarly a common factor is found to exist for any other case of parameters. We may therefore suspect the existence of a common factor when we find the equation of the secant in the general case. It is proposed to find this common factor and to apply the simplified equation to the solution of a well-known problem.

By a space of n dimensions we mean a region in which the position of a given point is determined by n independent coordinates. If the region is such that the shortest distance between any two given points is the straight line joining them, the space is termed flat, the geometry of a flat space is akin to that of the plane as contrasted with, say, spherical geometry For brevity, a flat space of n dimensions will be denoted by the symbol [n].

The relation between a curve and its involute suggests the idea of a pair of curves such that the binormals of one cut the other orthogonally. Let r be the position vector of the current point P on the first curve C and t, n, b the unit vectors in the directions of the tangent, the principal normal and the binormal at P These are functions of the arc-length s of the curve. Since the binormals of C intersect the second curve C1 the position vector of the point P1 of the second curve which lies on the binormal at P is given by

r1= r + ab,.....(1)

in which a is a scalar.

When the subject of the teaching of differentials was discussed by this Association at its Annual Meeting two years ago, I read a paper on the theoretical aspect of the subject. In that paper I maintained that it is futile to discuss the teaching of differentials until we have agreed upon the theory of differentials which we wish to teach, and I proceeded to summarise the three main theories of differentials. It was obvious that each of these theories was intrinsically more difficult and abstract than the theory of limiting processes which has become the classical method of introducing the calculus, and it was unanimously agreed that each theory was, in practice, quite unteachable.

The aim of the following notes is merely to indicate the straight-forward way in which vectors lend themselves to geometry. Some of the work is less direct than it might be, but this is partly due to the desire to keep to first principles as much as possible.

The chain of theorems in § 6 has been suggested by a chain constructed many years ago by Mr. H. W. Richmond. In this, points are taken on a unit circle and the second unit circle through each pair of points is drawn ; the centres give rise to a chain of unit circles.

1.1. Relation between mutual distances of 4 points in a plane.

Let O, A, B, C be the four given points.

In Hilton’s Plane Algebraic Curves the functions F(t) and F′(t) are used in finding inflexions and cusps of plane curves. These functions are defined, respectively, by the determinants

where f(t), ɸ(t), ψ(t) are the values, in terms of a parameter t, of the homogeneous coordinates of a point on the curve. It is shown that F(t1) vanishes if t1 is the parameter of an inflexion or a cusp and F′(t1), the derivative of F(t1), vanishes if t1 gives a cusp.

From the first inception of vector analysis, the subject of notation has engaged the attention of many writers, but up to the present there has, unfortunately, been no sign of general agreement upon a standard notation.

In a previous article (October 1933) it was shown how the “projection method” could be applied to the drawing of bending moment diagrams for fixed concentrated loads. It is proposed to extend this method to cases where there are uniformly distributed loads in addition to the concentrated ones.