“On Factors of Numbers of the Form , by Professor Sanjana, in Vol. XXVI. of the Proceedings.

Page 68, last line but one, for

Page 70, line 20, for “15 258 501” read “15 258 358 501.”

Page 74, line 2, for

Page 77, last line but one, for “4.1.744 287” read “4.1.7.44 287.”

Page 79, line 18, for “ products ” read “ product .”

Page 83, lines 6 and 8, for “ same ” read “ some.”

Page 84, footnote, for “ right ” read “ rigid.”

Let the convex quadrilateral formed by the four given tangents be ABA′B′, and O the intersection of the diagonals. Let OA and OB be taken as axes of x and y. Denote OA, OA′, OB and OB′ by a, a′, b and b′, a and b being positive, and a′ and b′ negative. The tangential equation of the system is then

where k is a variable parameter; for the equation is satisfied when the straight line lx + my + 1 = 0 passes through any two adjacent angular points of the quadrilateral.

The object of this paper is to essay an analytical statement of the reduction of the integration of a canonic system of differential equations (into which time does not enter explicitly) to that of the partial differential equation of Jacobi and Hamilton; and to illustrate the principle of duality by an outline of the solution for the problem of two bodies both by the standard form of the equation referred to and by the analogous form which that principle involves. Most statements of the reduction are verifications and somewhat obscure the symmetry of the canonic form. The shortest procedure, of course, is by means of the well known theorem of Jacobi, and this verificatory method is followed by Tisserand, Charlier and Appell. Poincaré gives a proof depending on a simple form given by him to the conditions for a canonical change of variables, but again the statement lacks analytical form. The essentials of this proof will be given here, but in an entirely different way. An analytical treatment of the subject has been given by Professor L. Becker in his class lectures at Glasgow, but it has not been published.

The anharmonic ratio of four points A, B, C, D depends on the order in which they are taken. The A.I.G.T. “Syllabus,” following Cremona, defines the symbol (ABCD) to stand for , when A is conjugate to B and C to D. Thus, taking λ to denote the value of this double-ratio, we have

The following is a problem in voting:

A number of n individuals are separately arranged in order of preference by a number of k electors ; it is desired to ascertain the order of preference in the opinion of the electors as a whole.

Without discussing what is the best solution of this problem, there are three plausible methods, and it will appear as the result of a curious theorem that two of them lead to identical results.

In the Cayley-Klein projective metric it is ordinarily assumed that the measure of angles, plane and dihedral, is always elliptic, i.e. in a sheaf of planes or lines there is no actual plane or line which makes an infinite angle with the others. With this restriction there are only three kinds of geometry—Parabolic, Hyperbolic and Elliptic, i.e. the geometries of Euclid, Lobachevskij and Riemann ; and the form of the absolute is also limited. Thus in plane geometry the only degenerate form of the absolute which is possible is two coincident straight lines and a pair of imaginary points ; in three dimensions the absolute cannot be a ruled quadric, other than two coincident planes. If, however, this restriction as to angular measurement is removed, there are 9 systems of plane geometry and 27 in three dimensions; for the measure of distance, plane angle and dihedral angle may be parabolic, hyperbolic, or elliptic.

The position of a point in a plane may be determined by its distances (r, ρ) from two fixed points which may be termed foci. These distances are termed vectorial coordinates. The determination is unique if attention is confined to the half-plane bounded by the line of foci. In certain cases where the properties of a curve are defined with respect to two points, representation of the curve by an equation between its vectorial coordinates possesses certain advantages. For example, the equation of the ellipse takes the extremely simple form r + p = 2α.

Let γn denote the value of

where n is a definite integer; and let γ denote the limit of

when the integer k is indefinitely increased. It is known that the expansion of γn – γ in ascending powers of 1/n is

where B1, B3, B5… are the numbers of Bernoulli. The series (3) is, however, divergent, as B2r+1 not only increases indefinitely with r, but bears† an infinite ratio to B2r–1 in this case. It is proposed to find by elementary methods the expansion of γn – γ up to the term in nr and to estimate the error (of order l/nr+1) made in omitting further terms of series (3). I shall take the case of r = 9, but the process is quite general.

The following note indicates how the equation to the locus of the straight lines which intersect three given lines may be obtained by the use of the conditions that three planes should have a line of intersection.

Three given planes

have a line of intersection if any two of the determinants in the scheme

are equal to zero.

f λS is the product of two factors of the form αx + βy + γz if λ is a root of a discriminating cubic

A well-known proof of the reality of the roots of the cubic is as follows :—

Write

In earlier volumes of the Proceedings of the Edinburgh Mathematical Society (cf. III., 104; IX., 83) it has been shown by researches of Dr. J. S. Mackay that the “discovery of the Wallace line … dates back only to about the year 1799 or 1800.” This result is reproduced in Cantor's Vorlesungen über die Geschichte der Mathematik (III., 542, 2te Aufl.). It was arrived at by considering the two following theorems given by Professor Wallace in the old series of Leybourn's Mathematical Repository:

Theorem A (Vol. I., p. 309 ; Vol. II., p. 54–5). If three straight lines touch a parabola, a circle described through their intersections shall pass through the focus of the parabola.

Non-Euclidean geometry in the narrowest sense is that system of geometry which is usually associated with the names of Lobachevskij and Bolyai, and which arose from the substitution for Euclid's parallel-postulate of a postulate admitting an infinity of lines through a fixed point not intersecting a given line, the two limits between the intersectors and the non-intersectors being called the parallels to the given line through the fixed point. In a wider sense, any system of geometry which denies one or more of the fundamental assumptions upon which Euclid's system is based is a non-euclidean geometry. Of special interest are, however, those which touch only the question of parallel lines ; and there exists, in addition to Lobachevskij's geometry, another, commonly associated with the name of Riemann, in which the parallels to any line through a fixed point are imaginary. The three geometries, Lobachevskij's, Euclid's, and Riemann's, thus form a trio characterised by the existence of real, coincident, or imaginary pairs of parallels through a given point to a given line. With reference to this criterion, a consistent nomenclature was introduced by Klein, who called these three geometries respectively Hyperbolic, Parabolic, and Elliptic.

1. Let the sides AB, AC subtend equal angles λ at P.

Take ABC as the triangle of reference and denote AP, BP, CP by x, y, z. Then we have, using trilinears,

In a recent paper read before the Society, Professor Carslaw gave an account, from the point of view of elementary geometry, of the well-known and beautiful concrete representation of hyperbolic geometry in which the non-Euclidean straight lines are represented by Euclidean circles which cut a fixed circle orthogonally. He also considered the case in which the fixed circle vanishes to a point, and showed that this corresponds to Euclidean geometry. The remaining case, in which the fixed circle is imaginary and which corresponds to elliptic or spherical geometry, is not open to the same elementary geometrical treatment, and Professor Carslaw therefore omitted any reference to it. As this might be misleading, the present note has been written primarily to supply this gap. It has been thought best, however, to give a short connected account of the whole matter from the foundation, from the point of view of analysis, omitting the detailed consequences which properly find a place in Professor Carslaw's paper.

Much has been written in recent years on the foundations of geometry, chiefly in Germany and Italy, and the relations of the various Non-Euclidean geometries to the Euclidean system are now more generally known among mathematicians. But most of these writings involve a knowledge of more advanced mathematics, while it has been found difficult to represent even the simplest Non-Euclidean geometry—that of Bolyai-Lobatschewsky—in an elementary manner.

The methods now adopted in the teaching of elementary geometry have made it most important that the teacher should have clear views upon the nature of the problems which are soluble by Euclid's methods: that is, with the aid of the ruler and compass only. With this general question I have dealt in another place. In this paper I give a short account of the argument by means of which Gauss proved that the only regular polygons of n sides, which can be constructed by Euclid's methods, are those in which n, when broken up into prime factors, takes the form

m1, m2, m3,…mr being all different.