[Of the following properties, some must have been long known, but I do not remember any early statement of them. The 1st, 10th, 15th, 16th, 19th, 20th, and 22nd are given, mostly without proof, in an article by Vecten in Gergonne's Annales de Matliématiques, vol. VII., p. 321 (1817); the first parts of the 3rd and 4th are proposed for proof by William Godward in The Gentleman's Diary for 1837, p. 48, and proved by him and others in the Diary for 1838, pp. 40-41; the first parts of the 5th, 6th, 12th, and 14th are given in M'Dowell's Exercises on Euclid and in Modern Geometry, §§ 27, 28, 29 (1863); the 9th in Milne's Weekly Problem Papers, p. 135 (1885); the first part of the 8th in Vuibert's Journal de Mathématiquts Elémentaires, 12° Année, p. 18 (1887); the 39th in the Journal de Mathématiques Elémentaires, edited by De Longchamps, 3° série,tome I.,p. 234 (1887). The others are believed to be new.

This is the problem of § 670 in Clerk Maxwell's Electricity and Magnetism. The author proposes to proceed by another method and to obtain the result in a different form. Let O be the centre of the spherical surface on which the shell lies and Z the point where the magnetic potential Vm is to be found. Also let ∅ be the strength of the shell (magnetic moment per unit area), α its internal, and α + δα its external radius. To represent the magnetic distribution let a layer of negative magnetic matter of density σ cover the inside face, and a corresponding positive layer the outside face. Finally, let Z be without the matter of the shell and on the positive side.

In spherical, as in plane geometry, transformation by inversion and by similar figures may be used for the deduction of new theorems from known ones, the two methods, however, in the former case becoming identical; and like all other propositions and methods in spherical geometry, these methods of transformation may be dualized. The transformation indicated as the dual of these in the case of the spherical surface is also applicable in piano. The sequel is an account of this method and of some results that may be obtained by means of it.

The approximation is

We have

Now assume

where A differs from 1/12 by a quantity of the order of 1/n, since the above series for x is convergent; and since n is large, A may be taken to be 1/12.

§ 1. The device here described has been found to simplify greatly the “somewhat laborious discussion” of the different musical intervals as given, say, in Sedley Taylor's Sound and Music (chap, viii.) or in Helmholtz's Sensations of Tone. It has been found particularly helpful in giving an account, necessarily rapid, of the nature of harmony to classes studying sound as a part of physics, from whom much familiarity with musical terms and notation is not to be expected.

§ 1. The object of this note is to establish the above inequality in as general a form as possible, and to prove by means of it two of the principal propositions in the theory of inequalities, one of which is usually proved by means of infinite series. The logical advantage in making the theory of inequalities independent of that of infinite series is obvious, when it is remarked that the discussion of the convergency of infinite series is strictly speaking a part of the theory of inequalities.

On donne deux cercles S et Σ, ayant pour centres les points O et ω, pour rayons r et ρ. Le cercle S est supposé intérieur au cercle Σ, et le point to intérieur au cercle S.

I. Tous les cercles T tangents extérieurement au cercle S et ortho gonaux au cercle Σ sont tangents à un troisième cercle fixe.

§ 1. To inscribe in a triangle ABC a triangle similar to the triangle DEF, and having its sides parallel to those of DEF.

In order to inscribe in the triangle ABO (fig. 21), a triangle having its sides parallel to those of DEF, through D, E, F, draw lines parallel to the sides of ABC, and then reduce the figure A′B′C′ in the ratio BC : B′C′.

[The present paper is a translation of the second part of the third book of Pappus's Mathematical Collection. Pappus's date is uncertain, but 300 a.d. may be taken as an approximation to it.

Throughout the translation I have used the word “progression” as a rendering of the Greek μεσότης, which has no English equivalent. The only other alternative was to employ the term mediety, from the Latin medietas.

The account of the various progressions given by Nicomachus, in his Arithmetical Introduction, differs somewhat from that of Pappus. I hope to have something to say about Nicomachus in a future paper.]

In a paper recently printed in the Society's Proceedings, I considered the effect of compressibility in the fluid on the motion of straight vortices; the present paper treats of circular vortex rings in a compressible fluid. The circle passing through the centres of the circular cross sections of the vortex filament will be called the “circular axis,” and the perpendicular to the plane of the circular axis through its centre, the “axis” of the vortex. In the notation employed, a denotes the radius of the circular axis, and e that of the cross section of the filament, while ω represents vorticity, and ρ density. It is also convenient to denote the area of the cross section— i.e., πe, by σ. Following Helmholtz, it will be supposed that e/a is always very small, and that the cross section is truly circular. Certain small inconsistencies in the ordinary theory following from this last assumption will be pointed out, though they do not seem seriously to affect the general applicability of the results. The axis of the vortex ring is taken as axis of z, and z, r, θ are the ordinary cylindrical co-ordinates. It is also convenient to denote by r' the distance of a point from the circular axis of a ring, and by ψ the inclination of this distance to the plane of the circular axis. The effects of the vorticity and variation in density may be considered separately.

The following paper contains little that can be regarded as new mathematical information. It aims only at showing, or rather at emphasising, the correspondence which exists between two geometrical theories which are related to each other in the same way as the arithmetical theories of multiplication and division. Such value, therefore, as it possesses is primarily pedagogical.

The following note may be considered as an addendum to the paper by me on pp. 42–47 of this volume of the Proceedings. In that paper it is shown how to inscribe in a triangle ABC, a triangle DEF, such that the perpendiculars to the sides of ABC, drawn through the points D, E, F, shall be concurrent in a point P. This is done by constructing on each of the sides of ABO a triangle similar to DEF; then O the point of concurrence of the three lines joining the vertices of ABC to the vertices of these triangles is the point “inverse” to P. The question, then, naturally arises, What must be the shape of the triangle DEF in order that the point P may be one of the Brocard points, and, as a consequence, O the other one? and the answer is easily seen to be that DEF must be similar to ABO. Hence the following construction:—