Theorem. If a point be taken on the radical axis of a coaxial system of circles, and from it tangents be drawn to any circle of the system, these tangents are cut in points on a conic, by the radical axis of the circle and a given fixed point. The two points are the foci of the conic. (Fig. 1.) Let W1W2 be the line of

gives an ellipse as the locus of P1P2, &c, when, as in Fig, 1, S is internal to F. If S were an external point, we should have P1F-P1S = P1F - P1f1 = radius of F = constant, and the locus of P1, P2, &c., would be a hyperbola. When F is at infinity on the radical axis, P1S = P1f1, and P1f1 being at right angles to W1W2, the conic is a parabola, and the line of centres the directrix.

1. In his Miscellanea Analytica of the year 1762, Waring stated the identities

where sp stands for the sum of the pth powers of n quantities α, β, γ,…. If the identities be written in the uncontracted form

it is readily seen that they hold not merely for certain powers of certain quantities, but for any quantities whatever; and that, indeed, for the elements of the array

we may legitimately substitute in order the elements of the array

thus obtaining the identities

These latter were given by Binet in 1812, and are usually, but with little real justification, spoken of as “Binet's Identities.”

The chain is supposed to be rotating bodily about a vertical axis with constant angular velocity, and to have taken up a shape of relative equilibrium; the links are like separate pendulums jointed together, and the condition is investigated where the chain is composed of one, two, three, or any number of such pendulum links; finally the passage is considered to the ultimate case of a chain of small links, which may be considered as a continuous flexible cord.

1. The following note examines, more closely than the text books usually do, some results of reciprocation, and has special reference to properties of a certain infinite set of triangles and connected conies.

1. The problem has been proposed by Steiner* of finding the envelope of a system of similar conics circumscribed about a given triangle, and of finding the loci of the centres and foci of the conics of the system. He states that the envelope is a curve of the fourth order having three double points, and gives some of its properties. The problem has been treated by P. H. Schoute in a paper entitled Application de la transformation par droites symétriques à un problème de Steiner.† In this paper the author discusses the problem of the envelope in detail by a geometrical method, and gives the order of the locus of centres, of the locus of foci, and of the locus of vertices, and the class of the envelope of asymptotes, of the envelope of axes, and of the envelope of directrices. But in every case (except that of the locus of centres), he gives the degree or class twice as great as it should be. In a paper published in the Annals of Mathematics‡ I have found explicit equations for the locus of centres and the envelope of asymptotes, showing that each asymptote envelopes a three-eusped hypocycloid; and in a paper in the Transactions of the American Mathematical Society,§ I have obtained an equation for the envelope of the axes, showing that each axis also envelopes a three-cusped hypocycloid. I have also obtained the equation of the envelope of the directrices (a curve of the fourth class) and the equation of the locus of the vertices (a curve of the eighth degree); but the results have not been published. In the solution of a problem in the Educational Times (proposed by himself), Cayley* proved that the locus of the foci of the parabolas which pass through three fixed points is a unicursal quintic passing through the two circular points at infinity, by showing that the co-ordinates of the focus of any such parabola may be explicitly expressed as rational functions of a parameter of the fifth degree. As the locus of the foci of a system of similar conies passing through three fixed points is not in general a unicursal curve, it is hardly likely that Cayley's method could be extended to the general case.

1. The six planes through the middle points of the edges of a tetrahedron perpendicular to the opposite edges are concurrent.

Let the equation of the four point system be

where h is a variable parameter and a, b, g, f and c are constants. The four fixed points are thus given by and . Since the quadrilateral is real g2>ac and f2>bc: it is convex if and have the same sign, i.e. if ab is positive, and concave if ab is negative.

1. Theorem I. If , where a and n are positive integers, and if f(n) is a finite single-valued function of n for values of n蠇a, then

The object of this note is to show that, in applying the methods of hypercomplex numbers to the theory of determinants, there is, for many purposes, no gain in using a particular number system.

The object of this note is to show that, in applying the methods of hypercomplex numbers to the theory of determinants, there is, for many purposes, no gain in using a particular number system.

The principal problem which we have in view may be stated as follows:

To construct a triangle ABC (Fig. 1) in which are given in magnitude only, the height h = AO from the vertex A, the median m = BI from the vertex B, and the bisector f= CD from the vertex C.

The principal problem which we have in view may be stated as follows:

To construct a triangle ABC (Fig. 1) in which are given in magnitude only, the height h = AO from the vertex A, the median m = BI from the vertex B, and the bisector f= CD from the vertex C.