1. If PQRS be an Hyperbola, OE, OF its asymptotes, P, Q, R, S any points on it such that the sectorial area OPQ = sectorial area ORS; and if PA, QB, RC, SD be ordinates to one asymptote and parallel to the other, it is known that

Hence if A, B, C … be taken so that OA, OB, OC … are in continued proportion, the areas OPQ, OQR, ORS … are all equal, and since the number of points can be made as large as we please, the sum of the sectorial areas can be made as large as we please. ∴ The area between an asymptote, the curve, and any radius vector is infinite.

With the issue of the Third Part of the Third Volume, Mr Cantor completes his History of Mathematics, in accordance with the plan he sketched out for himself when he undertook the work. That the labour involved in collecting material and in reducing it to shape would be great, Mr Cantor doubtless well knew; but in all probability his most liberal estimate of the demands likely to be made upon his energies has been far exceeded; in any case, one can readily understand the feelings of satisfaction with which he writes the preface to the concluding volume.

The object of the paper is to suggest a rearrangement of these theorems, which, it is contended, is simpler than, but as effective as, that usually adopted. The slightly more general theorems thus obtained suggest themselves readily, but the writer has not endeavoured to investigate other than the most elementary theorems in question.

In the proof of the binomial theorem for negative and fractional indices given in many text-books of algebra, and attributed to Euler, one step seems to me to involve a very gross assumption.

The symbol f(m) having been taken to denote the series

it is pointed out that whenever m and n are positive integers we know that f(x)×f(n)≡f(m + n); and the conclusion is drawn that since this is true for all positive integral values of m and n, by the “permanence of equivalent forms” (whatever that may mean) we can conclude that it is true also for negative and fractional values of m and n, whenever f(m) and f(n) are convergent.

Thus (1 + x)n − 1 is divisible by x.

Hence (1 + x)n = 1 + terms in x and its powers.

This is the problem discussed in my paper bearing the not very happy title “On the different non-linear arrangements of eight men on a chess-board”, which was read to the Edinburgh Mathematical Society on 14th March 1890, and is printed in its Transactions, Vol. VIII, p. 30. At that time I was not aware that the problem had been discussed by any previous writer, and I treated it as an entirely new one. I have since learnt that a good deal has been written about it, and I propose on the present occasion to give briefly the history of the problem, and the results which have been arrived at; also to communicate some new results which I have obtained.

The following is a simple geometrical demonstration of the well-known theorem that, if matter be distributed over a sphere with a surface-density (i.e., mass per unit of surface) inversely as the cube of the distance from either of two points which are the inversions of each other with respect to the sphere, it will act upon all external masses as if it were collected at the interior point:—and upon all internal masses as if a definite multiple of its mass were concentrated at the exterior point.

If a circle meets the sides BC, CA, AB of a triangle (Fig. 6) in the pairs of points L and l, M and m, N and n respectively, it is obvious that the pairs of connectors Mn and mN, Nl and nL, Lm and lM are antiparallel with respect to the angles A, B, C respectively.

It has been shown [4] that if two triangles ABC and PQR (Fig. 6) be in perspective with respect to any point S as centre of perspective, the sides of PQR cut the non-corresponding sides of ABC in three pairs of points which lie on a circle, provided that the angles made by the sides of PQR with those of ABC possess certain values dependent on the position of S. In the general case, when these angles do not conform to the condition referred to, it follows from Pascal's theorem that the six points determined by the sides of PQR on the sides of ABC, lie on a conic.

If A1 and B3 (Fig. 17), B1 and C3, C1 and D3, D1 and A3, A2 and C2, B2 and D2 be pairs of points on the edges

respectively of a tetrahedron ABCD, such that the two points on any edge are concyclic with the two points on any other edge—a manifestly possible condition of things—then the twelve points lie on a sphere.