The present note is an extension of a previous paper on the same subject. In this paper a concise proof was given of a theorem by Scheffers to the effect that if a linear associative algebra contains the quaternion algebra as a subalgebra, both having the same modulus, then it can be expressed as the direct product of that quaternion algebra and another algebra. It was also shown that this theorem could be generalised to the extent of substituting a matric quadrate algebra for the quaternion algebra. In the present paper the theorem is extended to certain other types of algebras.

1. If in a closed area CD a point O is given, and through this point a straight line MN is to be drawn such that the segment MCN so determined may be a minimum or a maximum, this condition is satisfied by drawing through O a chord MN having O for its mid point. Of the two segments which are separated by this chord and which are obtained by rotating the chord round O, the one MDN will be the maximum and the other MCN will be the minimum. The sum of the segments is equal to the whole area of the contour, and consequently if one of them corresponds to the minimum, the complementary segment corresponds to the maximum. The two segments interchange, the one into the other, when the chord MN, turning round the point O, undergoes a displacement equivalent to the angle π.

The locus of the focus of a parabola touching three given fixed straight lines is both exceedingly simple and widely known. On the other hand the analogous locus of the focus of a parabola passing through three given fixed points is excessively complicated, and its investigation has, so far as the present writer knows, never appeared in any text-book.

The integrals of §§ 5, 6, and 7 of the following paper were first established by C. de la Vallée Poussin in a memoir Sur quelques applications de l'intégrale de Poisson (Ann. de la Soc. sc. de Bruxelles, vol. 17, 1892–3). An analogous integral to that of § 5 was discovered by A. Hurwitz, who seems not to have been aware of de la Vallée Poussin's memoir, and will be found under the title Sur quelques applications des series de Fourier in the Annales de l'École normale, vol. 19, 1902. In view of the value of these integrals for the theory of the Fourier series, the discussion now given, which follows different lines from those of previous proofs, may be of some interest. The discussion turns chiefly on the Second Theorem of Mean Value which is quite as applicable to Poisson's as to Dirichlet's Integral.

1. The object of this paper is to correlate the geometrical and analytical aspects of the elements of the theory of points at infinity, etc., in a plane. It is assumed that the reader is acquainted with the method of tracing the graphs of rational functions of x, by using the artifices of change of origin and approximation by Ascending or Descending Division (see Chrystal's Introduction to Algebra, Ch. XXV.).

The following notation is here used for the quantities occurring in the discussion of tortuous curves.

Length = s; curvature = ; torsion = ; direction cosines of tangent, principal normal and binomial:—α, β, γ; l, m, n; λ, μ, ν.

Frenet's formulae are therefore,

with two corresponding sets.

Let a and b be positive and unequal; and let x be their arithmetic mean. Then we have the following equality and inequality

The inequality is tantamount to the fact that the G.M. of a and b is less than the A.M.

The following notes are intended to introduce a simple method of treating elementary geometrical conics, and at the same time to supply a missing link in the chain of continuity between Euclidean geometry and the modern methods of treating the conics, which at present are treated more as different subjects than as a continuous whole.

The substance of this paper is contained in Chrystal, chap, xxv., §§13, and 15 to 20, with some applications thereof occurring in chap. xxvi. But it is treated here in a fresh manner which would seem simpler on several points. This mode of presentation was, in the start, suggested by Peano's method given by Prof. Gibson in his “Note on the Fundamental Inequality Theorems Connected with ex and xm,” in Vol. XVIII. of the Proceedings.

Obscurity in the direct discussion of The Envelope, as given in works on Differential Equations, has led writers on The Calculus to define the envelope of a family by a property which all know that it shares with any locus of multiple-points belonging to the family. The following presentation is an attempt by use of systematic notation to make clear the details of the direct process:—

Starting from the definition that

A curve is an envelope of a given family, if at each of its points it touches a member of the family:

let us suppose that a family is specified by the equation

in which ψ is a continuous function of the three variables x, y, u; continuous variation of u corresponds to continuous motion and deformation of a variable curve in the xy-plane, which takes in succession the curves of the family as positions.

1. In the Cambridge and Dublin Mathematical Journal, vol. v., 1859, De Morgan gives the definition of the “area contained within a circuit” as the area swept out by a radius vector which has one end (the pole) fixed and the other describing the circuit (in a determinate mode), on the supposition that each element of area is positive or negative, according as the radius is revolving positively or negatively. He remarks that the definition satisfies existing notions, that it provides the necessary extension of the meaning of the word area, and proceeds to show that it gives to every circuit the same area, whatever point the pole may be. The object of this paper is to give an Area-Theory beginning with the triangle and going on to circuits bounded by straight or curved lines. The fundamental proposition is derived from Analysis, and the geometry of the applications is therefore an Analytical Geometry; indeed, one of the objects of the paper is to emphasise the advantage of keeping Analysis and Geometry in close correspondence. As evidence of the difficulty of pursuing an Area-Theory in Geometry, without the aid of Analysis, it may be noticed that Townsend in his Modern Geometry (1863), §83, lays down Salmon's Theorem in this form: “If A, B, C, D be any four points on a circle taken in the order of their disposition, and P any fifth point, without, within, or upon the circle, but not at infinity, then always area BCD.AP2 − area CDA.BP2 + area DAB.CP2 − area ABC.DP2 = 0, regard being had only to the absolute magnitudes of the several areas which from their disposition are incapable of being compared in sign.”

In the first part of this paper there are found the numbers of points, lines, etc., in a finite projective geometry of n dimensions. The substance of this has already been worked out by O. Veblen and W. H. Bussey. The second part is concerned with the arrangements of the numbers representing the points in a finite projective plane desarguesian geometry.

The square described on the hypotenuse of a right-angled triangle is equal to the squares described on the other two sides.

About half a hundred proofs of this theorem have been given, but few of them have been “ocular,” that is, few have shown how the two smaller squares may be decomposed so as to fit into the largest square. One of the most elegant of the ocular proofs is that of Henry Perigal, and was discovered about 1830. A demonstration of its correctness is not difficult to obtain, but the following demonstration is believed to be new. It depends somewhat on algebra, and presupposes a simple lemma.

About forty years ago, when Spencer was rising into philosophic fame, it used often to be said by his admirers that he was an accomplished mathematician. This statement was accepted without demur, though it was known that he had not measured himself against rivals of his own age, or, what is more important, had not produced anything new in this old science.

1. The term Second Moment, which is already in frequent use, as applied to lines, areas and volumes, as well as masses, is preferable to the older term Moment of Inertia which properly applies only to masses.