It is very seldom that any properties of a pencil of cubics can be discovered beyond some very general ones. The above case yields, however, some interesting results. It presented itself to me when I was considering the apolar generation of cubic curves which was published in the Proceedings of the London Mathematical Society (Ser. 2, Vol. 9, Part 3), and which I shall have frequent occasion to refer to in the present paper.

An infinite family of triangles, having a common pole (determined by three fixed lines through it), and polar (determined by three fixed points on it), and an allied family of conics with imaginary double contact.

Construction for pole of a line with reference to a triangle.

One of the most plausible of the host of “proofs” that have ever been offered for Euclid's parallel-postulate is that known as Bertrand's, which is based upon a consideration of infinite areas. The area of the whole plane being regarded as an infinity of the second order, the area of a strip of plane surface bounded by a linear segment AB and the rays AA′, BB perpendicular to AB is an infinity of the first order, since a single infinity of such strips is required to cover the plane. On the other hand, the area contained between two intersecting straight lines is an infinity of the same order as the plane, since the plane can be covered by a finite number of such sectors. Hence if AP is drawn making any angle, however small, with AA′, the area A′AP, an infinity of the second order, cannot be contained within the area A′ABB′, an infinity of the first order, and therefore AP must cut BB′. And this is just Euclid's postulate.

Cayley has shown that if the family of surfaces φ(x, y, z) = λ is one of an orthogonal triad, φ satisfies a differential equation of the third order. If, however, the parameter λ is involved implicitly in the equation of the family, the condition requires modification: for example, although

is one of an orthogonal triad, φ does not satisfy Cayley's equation.

I propose in this paper to investigate the geometrical nature of a certain species of differential equation, which includes, as a particular case, Clairaut's well worn equation

The theory of graphical integration is founded on the graphical integration of parabolic arcs. The details and the different technical applications of these problems are developed in the fundamental work of J. Massau; we find there very simple pure constructive methods for the case of parabolas till the 3rd order.

In the Appendix III. of his “Mechanics of Machinery,” Le Conte attacks the problem of rolling curves by an elegant analytical method in which the cartesian forms of the involute and the trochoids are derived from elementary differential equations. Weisbach(“Mechanics of Engineering and of Machinery,” Vol. III., chap. II.), while using mainly the geometrical method, discusses one application analytically (see §3 of this paper) in which polar forms are introduced. Many other writers, for example Barr (“Kinematics of Machinery,” chaps. III., IV.), deal wholly with the geometry of the subject.

The important relation in the Theory of Bending between the curves of Bending Moment (B.M.), Shearing Force (S.F.), and Load, or between those of Deflection, Slope, and Bending Moment, viz., that the tangents to the first of either set intersect in a vertical line through the centroid of the corresponding area of the last, under the usual convention of drawing, is usually not proved in Engineering Treatises, or else is established in simple cases by the polygon of loads.

The following is an easy method of representing geometrically the Sextic Covariant of a Binary Quartic which I have not seen given elsewhere. It was suggested to me by my work on Bi-Circular Quartics. The usual method of representing this Covariant is to regard the Quartic as four points on a conic, when the Sextic is represented by the intersections of the conic and the sides of the Harmonic Triangle of the Quadrangle whose vertices are the points of the given Quartic. The present method has the advantage of regarding the given Binary Quartic as a Tetrad of points on a straight line.

The algebraic methods employed are usually defective in explaining the sign to be attached to an area; while the trigonometrical method with the transformation of a trigonometrical formula has always seemed a little far fetched so early in analytical geometry.

The proposed mechanism explained here is based on the geometrical properties of Fig. 1.

0ABO′ is a straight line having 0A = BO′ = 1, and OB = AO′ = x and AH, BK are lines perpendicular to AB.

Last session (1910–11) I contributed two papers investigating the Focal properties of Circular Cubics and Bi-Circular Quartics by the methods of pure geometry. I find on further investigation that not only can the properties of the individual Focal Circles be directly found by the method there given, but the mutual properties of the Focal Circles can be obtained by an extension of the same method. Several geometrical results, which as far as I know are new, are obtained incidentally in the discussion.

In the works known to us which contain historical or bibliographical information about circular cubics no sufficient indication of the researches of Maclaurin is found. Yet many of the classic propositions connected with these curves are due to this eminent geometer, who investigated constructions for unicursal circular cubics, for certain non-unicursal circular cubics, and for the special circular cubics now known as the trisectrix of Maclaurin, the oblique cissoid, and the strophoid. His name does not appear even in the list of writings on the strophoid published by Tortolini and Günther.

In the first section of Kneser's book on Integral Equations and their Applications to Mathematical Physics, he applies that theory to the solution of some of the problems which arise in the Linear Flow of Heat. The object of this paper is to illustrate Kneser's use of Integral Equations in the Mathematical Theory of the Conduction of Heat by the discussion of one of the classical problems of Linear Flow which he leaves untouched.

Theorem I. Let nαp denote

where Then for any change in the independent variable x,

say z = f(x), the coefficient of in is nαp.

The main points of this paper are :—

(i) The construction of a linear substitution from its poles or linear invariants and its multipliers (§§3, 7);

(ii) a formula for r repetitions of a substitution (§ 4);

(iii) a specification of the types of linear substitutions of order r, with examples of the simplest of those types(§§9ff.);

(iv) The geometrical illustration of the case of three variables (§§15 ff.)