The generators and their orthogonal trajectories form, perhaps, the most useful set of parametric curves for the study of the local geometry of a ruled surface. It is not generally realised, however, that the fundamental quantities of the surface can be expressed quite simply in terms of the geodesic curvature, the geodesic torsion and the normal curvature of the directrix, that particular orthogonal trajectory which is chosen as base curve. Certain of the results are similar in form to those arising in the special case of a surface which is generated by the principal normals to a given curve, except that the curvature and torsion are geodetic. In addition it is possible to obtain in an elegant form the differential equation of the curved asymptotic lines and the expression for the mean curvature.

I have constructed two simple proofs of the result established by J. C. P. Miller, Mathematical Notes, No. 29 (1935), pp. vi–viii.

In geometry of three dimensions it is well known that, when two quadrics Q1, Q2 are given, if one set of four points exists having the properties that each point lies on Q1, and each two points are conjugate with respect to Q2, an infinity of such sets of points can be found. The quadrics Q1Q2 stand in a special relation to one another, expressed by the vanishing of the coefficient of A in the discriminant of Q2 + λQ1 an invariant of Q1, Q2. Two quadrics Q1, Q2 are thus related if the equation of Q1 contains no squares of the coordinates, and that of Q2 contains no products of two coordinates; for then the vertices of the tetrahedron of reference form such a set of four points.

1. If the approximate numerical value of e is expressed as a continued fraction the result is

and it was in finding the proof that the sequence extends correctly to infinity that the following work was done. First the continued fraction may be simplified by setting down the difference equations for numerator and denominator as usual, and eliminating two out of every successive three equations. A difference equation is thus formed between the first, fourth, seventh, tenth … convergents , and this equation will generate another continued fraction. After a little rearrangement of the first two members it appears that (1) implies

2. We therefore consider the continued fraction

which includes (2), and also certain continued fractions which were discussed by Prof. Turnbull. He evaluated them without solving the difference equations, and it is the purpose here to show how the difference equations may be solved completely both in his cases and in the different problem of (2). It will appear that the work is connected with certain types of hypergeometric function, but I shall not go into this deeply.

The position of the centre S of spherical curvature at a point P of a given curve C may be found in the following manner, regarding S as the limiting position of the centre of a sphere through four adjacent points P, P1, P2, P3 on the curve, as these points tend to coincidence at P. The centre of a sphere through P and P1 lies on the plane which is the perpendicular bisector of the chord PP1 and so on. Thus the centre of spherical curvature is the limiting position of the intersection of three normal planes at adjacent points. Let s be the arc-length of the curve C, r the position vector of the point P, and t, n, b unit vectors in the directions of the tangent, principal normal and binormal at P. Then if s is the position vector of the current point on the normal plane at P, the equation of this plane is

Since r and t are functions of s, the limiting position of the line of intersection of the normal planes at P and an adjacent point (i.e. the polar line) is determined by (1) and the equation obtained by differentiating this with respect to s, viz.

In the last few years the topic of differentials has occupied the attention of a good many teachers of mathematics. Since the appearance in 1931 of my article advocating the teaching of the Differential Calculus from the differential standpoint right from the start, very divergent views have been expressed on this question and there was a general discussion on the topic at the annual meeting of the Mathematical Association in 1934. This article, the subsequent correspondence and the discussion have certainly brought into evidence the fact that many mathematicians still have a very obvious distrust of “differentials” and they seem unable to clear their minds of some early prejudice against the employment of differentials for any purpose.

Suppose a polynomial or convergent power series

is raised to powers j = 0, 1, 2, 3, … The coefficients of xk in [f(x)]j, k = 0, 1, 2, …, may be entered as elements in positions (j, k) in an array or matrix F, thus:

By construction all elements in column (k) have weight (sum of suffixes) equal to k.

The important Binet-Cauchy theorem of determinants (3, p. 81)

may be illustrated as follows. Consider the identity

where ax = a1x1 + a2x2 + a3x3, and similarly for ay, bx, by. This identity follows by adding x1 row1 + x2row2 + x3row3 to row4 in the first determinant; and y1row1 + etc. to row5. Then by expanding both determinants in a Laplace development of the first three rows and the last two we obtain the identity

.

There are some people who find it easier to absorb abstruse theories when encouraged by picturesque analogies. There are also some people, mathematicians, who hold such assistance in austere contempt. Pictorialism, however, is no new thing in expositions of relativity, and I will not apologise for the following attempt to introduce a little more. The object of this note is to suggest ways of visualising some of the metrics which are of importance in the general theory of relativity. For each metric two pictures are supplied; they may be called (A) geometrical and (B) dynamical.