The most important application of the rational point transformations between two spaces lies in the construction of algebraical surfaces possessing singularities of various kinds and the investigation of their properties.

When a plane is transformed by such a transformation into a rational algebraical surface (homaloid) the geometry of the surface is immediately derivable from the geometry of the plane.

In the paper, “Transformations founded on the Twisted Cubic and its Chord System,” a series of space transformations was described, each of which had the property of transforming the chord system of one cubic into the chord system of the other. In the present paper it is shown that by the aid of a non-singular cubic surface the transformations of orders 1, 2, 3, 4, 5 may be derived directly without the intervention of a space transformation.

It will be found that these transformations form a group which is intimately associated with the 27 lines of the cubic.

While engaged in a study of the Methodus Differentialis of Jas. Stirling (1730) I have been struck by the fact that Nicole's Papers on the same subject, printed in the Memoires de l'Academie Royale des Sciences (Paris), appear to form a fitting prelude to the work published by Stirling. The dates of Nicole's Papers are 1717, 1723, 1724, 1727, and it is almost certain that Stirling was well acquainted with their contents, for he remarks on page 24 of the Methodus Differentialis:—“Hac de re primus quod sciarn egit D. Taylor in Methodo Incrementorum. Eadem etiam fusius et elegantissime traditur a D. Nicol in Actis Academiae Regiae Parisiensis.”

In 1730 there was published Stirling's Methodus Differentialis, and in it (Prop. VIII., p. 44) he considers the Difference Equation

and shows that it is satisfied by an inverse factorial series

The application of the geometrical properties of the Brocard and Tucker circles of a triangle to a quadrilateral appears never to have been adequately worked out, as far as the author can discover. Hence, the object of this paper.

Some of the problems involved have been published, under the author's name, as independent questions for solution, and where, in the author's opinion, solutions other than his own have seemed more satisfactory for the logical treatment of the subject, these solutions have been employed, with due acknowledgments to their authors.

In Vol. XXXV. (Session 1916–17), Part I., of the Proceedings of the Edinburgh Mathematical Society, I discussed in considerable detail the properties of the Apolar Locus of two tetrads of points. I showed there that, subject to certain defined conditions, a unique quartic curve would be obtained, which would be the Apolar Locus of the two given tetrads. I mentioned, however, in §7 of the paper, that in the case when the two tetrads lie on the same conic, the above-mentioned conditions are not independent, and that, in fact, not a unique quartic but a pencil of quartics is obtained.

The object of this note is to draw attention to a simple extension of a well-known theorem concerning corresponding points on confocal ellipsoids–an extension which seems to have escaped notice. It has been familiar to me for years as an illustration of the quaternion treatment.

The length of an arc of a flexible rope or chain suspended in the catenary y = c cosh x/c is s = c sinh x/c when measured from the vertex, but the practical determination of s is troublesome, owing to the difficulty in finding the parameter c from the transcendental equation when the coordinates of the point of suspension are given. The importance of a formula such as Huygens' approximation to the length of a circular arc s = 2B+⅓(2B−A), where A is the chord of the arc and B that of half the arc, is due to the fact that one can scale directly these lengths by rectilinear measurements without requiring to find the central angle or to make any subsidiary calculations. Formulae of this nature applicable to the parabola, or to curves whose arcs might be replaced by parabolic arcs, would be useful in the design of structural works dealing with ropes or chains. In such cases, as the dip is frequently less than one-eighth of the space, the catenary may be replaced by a parabola.

For determining the greatest bending moment produced by a rolling load consisting of two concentrated loads at a fixed distance from one another passing over a span, the usual method involves somewhat intricate algebraic discussion. Even for two concentrated loads the discussion is complicated, and for three it would be still more so.

The method here explained, which applies directly to the case when the whole of the rolling load is on the span, uses the shearing force curve only, and gives a very simple geometrical construction for finding the position and magnitude of the greatest B.M.

We will begin by stating the construction, and then proceed to give a geometrical proof.

Statement of the formula and numerical examples of it.

The object of the present note is to communicate the following formula for the solution of algebraic or transcendental equations:

The root of the equation

The present paper is concerned with determinants whose elements are themselves determinants. The best-known determinants of this kind are those whose elements are minors of a given determinant; these are called the “adjugate” and the “compounds” of the given determinant. Determinants whose elements are themselves determinants also occur frequently in Muir's theory of Extensionals.

The determinants considered in the present paper are of a somewhat more general type than adjugates, compounds, and extensionals; the principal result obtained is “Theorem A” (at the end of § 2), which relates to determinants whose elements are formed from any number of arrays. It is shown in § 3 that many other formulae, both new and old, may be obtained by specialising the arrays in “Theorem A.”