Def. The arithmetic mean of n quantities a, b, c, d …

Def. The geometric mean of n quantities a, b, c, d …

The problem discussed is that of dissecting two given triangles into triangular parts which shall consist of mutually similar pairs of triangles, so that the first given triangle A being dissected into the triangles a1, a2, a3 …, and the second given triangle B being dissected into the triangles b1, b2, b3 …, we shall have a1 similar to b1, a2 to b2, and so on.

The transformations discussed in the present paper are, like the isogonal and isotomic transformations, particular cases of the general birational quadratic transformation, in which points correspond to points, and lines to conics passing through three fixed points. They seem to possess some interest in connection with the Geometry of the Triangle.

In a paper by Mr Arthur Berry, M.A., in the Proceedings of the Cambridge Philosophical Society, Volume X. Pt. I., “On the Evaluation of a certain Determinant which occurs in the mathematical theory of statistics and in that of elliptic geometry of any number of dimensions,” a remark is made that in the case of n = 3 this determinant was readily evaluated by me by means of the formulæ of spherical trigonometry. I have thought that it might be of interest to show this evaluation, but I shall merely state the determinant at once of order 3, and leave the reader to refer to the paper quoted for the general determinant.

This note is intended to be supplementary to the paper, by Mr Muirhead, on “The dissection of any two triangles into mutually similar pairs of triangles.” The constructions given there, for the general case of this problem, yield no real solution if one angle of one triangle be greater than the sum of any two angles of the other triangle. For this particular case, the following constructions supply the necessary requirements; the first leads to a division of the triangles into three parts, the second to a division into four parts.

Let P be the point at which the potential has to be found (Fig. 7). Let the uniform surface density be σ, and take two radii PQ, PQ′ differing in length by a small quantity Q′K. Let QR be drawn perpendicular to BD, the diameter through P, and let CL be drawn perpendicular to PQ.

On reading Dr Peddie's paper, the following modification of the proof, which avoids summation, occurred to me:—

If in Figure 7 we take a point S on the circle BQD such that PQ + PS = 2a, where a is the radius, and a corresponding point S′ such that PQ′ + PS′ = 2a, then it is clear by Dr Peddie's construction that the potential at P due to the zone of the spherical surface lying between planes through Q and Q′ perpendicular to BD is given by 2πσ(PQ′−PQ) · a/CP, and is therefore equal to that due to the corresponding zone between S and S′, since

In the following paper, the determination of Green's function, for spaces bounded by surfaces of the cylindrical and spherical polar systems, is effected by what is believed to be a novel process, in which are utilised the properties of cylindrical and spherical harmonics, regarded as functions of their parameters.

The subject of this note is that dealt with in Mr Tweedie's paper in the Proceedings, vol. XVII., 33–37, and my only reason for bringing it before the Society is to call attention to a slightly different method of presenting the same order of ideas. The method is that adopted by Peano, Lezioni di Analisi Infinitesimale, vol. I., §23, but as the book is not readily accessible to teachers, there may be some interest in having the method reproduced in our Proceedings. I add one or two remarks.

Etant donne un triangle OAB, on demande de mener par le sommet O une droite OM telle qu' en abaissant les perpendiculaires Aa, Bb sur cette droite, les surfaces des triangles OAa, OBb soient entre elles dans un rapport r donné, c'est-à-dire qu'on ait la relation

The difficulty that is often felt by beginners as to the generality of the proof by projection of the addition theorem in trigonometry and of the formulae for change of axes in coordinate geometry seems to me to be due chiefly to the incomplete treatment of the fundamental theorems on projection. I do not mean that the proofs in the text-books are inaccurate, but only that sufficient pains are not always taken to show how the projection of a directed segment is to be measured so that the generality of subsequent proofs shall be beyond all question.