The chief purpose of the paper was to indicate the rise of transformations of the type where the integral in the first member of the equation is taken throughout a closed surface, and that in the second member over the surface, a being the angle made with the axis of x by the normal to the element dS drawn outwards. It is on this transformation the analytical proof of Green's theorem depends, and it was shown to have been employed in various forms by Poisson, Duhamel, Gauss, and others, before Green's essay was generally known on the Continent. It may be observed that the essay was published at Nottingham in 1828, and seems to have been unknown to continental mathematicians till its reprint in Crelle's Journal, vols. 39 (1850), 44 (1852), and 47 (1854).

The questions involved in the consideration of three-bar motion have attracted a good deal of attention (Proceedings of Mathematical Society of London passim, and elsewhere); but I am not aware of any complete account of the figures that can be derived from such a motion. The present paper gives a complete list of all the different kinds of curve that are obtained by a tracing point at the middle of the middle bar, the two outer bars being equal.

The number of groups of n which may be selected from 2n is 2n(2n − l)…(n + l)/n! But make the 2n into two groups of n, and select r out of the first and n − r out of the second. This gives [n(n−l)…(n−r+l)/r!] + [n(n−l)…(r + l)/(n−r)!] ways of thus making a group of n. Hence

1. The following condition may be new; it does not appear in any of the books:—

The condition that the straight line

should be a normal to the general conic,

referred to rectangular axes, is

where Σ is written for

and Δ, A, B, C, F, G, H have their usual meanings.

The idea in the following note is evidently capable of very wide development, but it can be made clear by a very simple example.

Whatever be the vectors α, β, γ, δ, we have always

But vector operators are to be treated in all respects like vectors, provided each be always kept before its subject.

The property referred to comes to light on consideration of the problem, “To inscribe in a given n-gon the n-gon of minimum perimeter.”

§1. In a triangle ABC (fig. 37), BE is made equal to CF; to find the locus of the middle point of EF.

Take K the middle point of BC and P the middle point of EF, then PK is the locus required. For if E′ and F′ are the middle points of BE and CF, the middle point of E′F′ will lie in PK (namely, at the middle point of PK); and again if BE′ and CF′ are bisected in E″ and F″, the middle point of E″F″ will lie in PK (namely, at the middle point of KR); and so on. At any stage we may double the parts cut off from BA and CA instead of bisecting them. Hence the locus required is such that any part of it, however small, contains an infinite number of collinear points; and hence the locus is a straight line.

The question having been proposed to me as a puzzle: To arrange eight men on a chess-board, so that no two of them shall be in the same line,—that is to say, that no two are to be in the same column, nor in the same row, nor in the same diagonal line,—I succeeded before very long in solving it by finding the annexed arrangement. (Fig. 45.)

In several previous communications to the Society, I have considered the equations of vortex motion in two dimensions in a compressible fluid. In the present communication I propose to consider certain forms of the hydro-dynamical equations of a more general kind. In certain cases the fluid will be supposed to be rotating, prior to the introduction of the vortex motion, with uniform angular velocity about a fixed axis.

The problem to be considered may be stated as follows:—How many necklaces may be formed with p pearls, r rubies, and d diamonds?

Giffard's injector appeared more than thirty years ago. The first serious attempt to explain its action on dynamical principles was made by the late William Froude at the Oxford Meeting of the British Association in 1860. The history of mechanical science is almost everywhere deeply marked by Rankine; and it seems, just as it ought to be, that he should be found to have contributed not a little to the literature of this particular subject in a paper presented to the Royal Society of London in 1870. As serving to show how far the problem is still interesting, even from a high standpoint, attention may be directed to the exceedingly curious procedure of Professor Greenhill, where he deals cursorily with the matter at the page numbered 448 of his article on Hydromechanics in the Encyclopædia Britannica.

In a short course of lectures on “Railway Practice,” the author was recently called upon to deal with the mechanical principles involved in the design of retaining walls. What has come to be known as Rankine's method had to be explained, at all events, in its practical application. But the time at the author's disposal did not admit of the general consideration of the theory of stress by which Rankine in characteristic fashion leads up (in his Applied Mechanics) to the particular problem under discussion. The author had therefore to choose between omitting a demonstration or making a short cut, which at the same time should be of a rigorous nature.

The object of this paper is to show how the leading properties of Spherical Harmonic Functions may be readily deduced by employing as a typical harmonic a certain simple algebraical expression, which obviously satisfies Laplace's equation; and to extend a similar method to the case of any number of variables.

The object of the paper was to suggest for the teaching of electrostatics a leading idea, which should readily co-ordinate all the facts, introduce no misleading inferences, and guide the course of learners in the direction of the most recent investigations—in all which respects the notion of attraction and repulsion is at least a partial failure. The leading idea or fact referred to is, that almost all electrostatic distributions, however complex, can be analysed into one or more repetitions of a certain simple system, which is called in the paper “an electrostatic system,” and which may be described as follows:—Two equally and oppositely electrified conducting surfaces, facing each other, separated by any dielectric, and insulated from each other. A complete study of one system of this kind, and of the very simple ways in which the establishment of one such system often necessitates the establishment of others, is therefore the fundamental study of electrostatics.

“The following is, of course, substantially well-known, but it strikes me as rather pretty:—to find the Orthomorphic transformation of the circle

into itself. Assume this to be

§1. The object of this note is, in the first place, to show that Menelaus's Theorem, regarding the segments into which the sides of a triangle are divided by any transversal, is a particular form of the condition, in trilinear co-ordinates, for the collinearity of three points; and, in the second place, to point out an analogue of Menelaus's Theorem in space of three dimensions.