1. The notation for a rectangular array can be extended so as to admit of arrays in which the number of rows exceeds the number of columns.

Let

denote the aggregate of all determinants of the mth order which can be formed from the rectangular array of pq elements by deleting p – m columns and q – m rows.

§ 1. In the Life and Letters of Lewis Carroll occurs the following extract from his Diary:—

“Dec. 19 (Sun).—Sat up last night till 4 a.m., over a tempting problem, sent me from New York, ‘to find 3 equal rational-sided rt.-angled ∆&s.’ I found two, whose sides are 20, 21, 29; 12, 35, 37; but could not find three.' (v. page 343.)

§ 1. The roots of the cubic equation

are given by the formula

This solution is of little practical use when the roots of the cubic are all real and unequal, that is, when is negative (the-Irreducible Case of Cardan's Solution).

1. The Pole and Polar Theorem for the Circle may be proved as follows:—

Let P be any point, and C the centre of a circle.

Let AB be the diameter through P, and QQ′ any chord passing through P.

Let AQ, BQ′ meet in M; AQ′, BQ in O; MO, AB in N; and MO, QQ′ in P′.

Mr H. Poincaré, in a recent work (la Valeur de la Science) speaks of the interest attaching to proofs of theorems in pure geometry by arguments derived from mechanical or physical considerations. The following is a case in point.

To find the centre of gravity of a circular arc by reference to a principle of dynamics.

If we join the angular points ABC of a triangle to any point O the locus of the centres of conies passing through A, B, C, O is a conic bisecting the six joins of the four points and passing through the intersections of OA, BC : OB, AC; and OC, AB. This conic is analogous to the nine-point circle, and at last meeting of the Society Mr Pinkerton showed that its centre lies on the line joining O to the centroid. In what follows an attempt is made still further to generalise this conception.

ABCDEF is any cyclic hexagon;

AB, DE meet in G,

BC, EF „ „ H,

CD, FA „ „ K; then G, H, K are in a straight line.

Draw KLNM parallel to BC and produce DE, EF, AB to meet it in I, N, M. Join DA, DM and let BC, DE meet in P.

§ 1. The inequality of the Arithmetic and Geometric Means of n positive quantities has been proved by many different methods; of which a classified summary has been given in the Mathematical Gazette (Vol. II., p. 283). The present article may be looked on as supplementary to that summary. It deals with proofs that belong to a general type, of which the proof given in the Tutorial Algebra, §205, and that given by Mr G. E. Crawford in our Proceedings, Vol. XVIII., p. 2, are very special limiting cases. Proofs of the type in question consist of a finite number of steps, by which, starting from the n given quantities, and changing two at a time according to some law, we reach a new set of quantities whose arithmetic mean is not greater, and whose geometric mean is not less than the corresponding means of the given quantities.

The condition that the principal normals of one curve may also be the principal normals of a second curve is, as found by Bertrand, that a linear relation with constant coefficients should exist between the curvature and torsion of each curve. In seeking for pairs of curves such that the tangents, principal normals or binomials of one may be the tangents, principal normals or binormals of the other, there are six cases to be considered. The curves of Bertrand are furnished by one case, and a second case, that of evolutes and involutes, is also discussed in the text-books. Of the remaining four only one gives results worthy of mention. Bertrand's problem suggests the inquiry into the nature of the pair of curves when the binormal of one is the principal normal of the other. A certain quadratic relation of a simple character found to exist between the curvature and torsion of the second curve led me to a paper in the Comples Rendus of 1893, by Demoulin, in which the problem had been generalised. His method of solution is different, and no explicit results as to the nature of the curves are given in the paper. Since no indication of the discussion of the problem is given in the text-books I have seen, I venture to submit a note of some results.

The homogeneous real linear transformation in n variables is such that, when these variables are used as a set of mutually rectangular coordinates, an n-dimensional sphere is transformed into an n-dimensional ellipsoid; n mutually rectangular radii of the sphere become the n, mutually rectangular, principal radii of the ellipsoid. When these principal radii have not been rotated from their original directions, the transformation is said to be pure, or irrotational. Since these radii are necessarily real, the roots of the n-ic for the determination of the n principal elongations are necessarily real.

Let S be the focus of the conic, PX the directrix, e the eccentricity.

Let V be the middle point of PP′.

Draw VK perpendicular to the directrix, and with centre V describe a circle, radius equal to e.VK.

Join SP, SP′ and draw radii Vp, Vp′, parallel to SP′, SP, and let PP′ meet the directrix in L. Then p, p′ are on the line SL.

1. By a quantitative proof of an inequality I mean one which exhibits the difference between the two magnitudes compared in a form which shows at a glance whether the difference is positive or negative. Such a proof not merely establishes the existence of the inequality, but also gives a measure of its amount.

1. This paper contains a number of investigations, more or less connected, on the theory of systems of circles. In such a well-worn field one does not expect to have hit upon much that is absolutely new, but it may be hoped that there is sufficient freshness of treatment to give the paper some interest even where it deals with results already known.