It is known to all students of physics that when a number of different observations have been made of the same constant quantity, the best result is, in general, obtained by taking the arithmetic mean of the observed values.

If the observed quantity instead of being constant is increasing at a uniform rate, and we require to determine this rate of increase, we should naturally expect that the proper course to follow would be to find the rates of increase of the quantity in successive equal intervals of time and take their arithmetic mean.

From this moment a brilliant period opens for geometrical research of every kind. Analysts interpret all their results, and set to work to translate them by constructions. Geometers endeavour to discover in every question some general principle—in most cases impossible to prove without the aid of Analysis—so as to deduce a host of particular consequences, closely connected with each other and with the principle from which they are derived. Jacobi’s brilliant pupil, Otto Hesse, admirably develops the methods of homogeneous coordinates of which Plücker, perhaps, did not fully appreciate the value. Boole discovers in Bobillier’s polars the first notion of a covariant ; the theory of forms is created by the labours of Cayley, Sylvester, Hermite, and Brioschi. Aronhold, Clebsch, Gordan, and other geometers still with us, invent the definitive notation of the theory, establish the fundamental theorem relative to the limitation of the number of covariant forms, and thus succeed in giving to the theory its fullest extension and scope.

The theory of surfaces of the second order, constructed mainly by the school of Monge, is enriched by a large number of elegant properties—mostly due to Hesse, who later finds in Paul Serret a worthy rival and an investigator who will continue his work.

Without restricting ourselves to the historical order, let us resume the development of mathematical physics in the last century, in so far as analysis is concerned. The problems of thermal equilibrium lead to the equation already known to Laplace in the study of attraction. There are few equations which have been the object of so many researches as this celebrated one. The conditions for the limits may assume various forms. The simplest case is that of the thermal equilibrium of a body, the elements of the surface of which are maintained at given temperatures. From the physical point of view, it may be regarded as evident that the temperature, assumed continuous in the interior since there is no source of heat, is determined when it is given at the surface. The more general case is that in which, the condition remaining permanent, there would be a radiation outward with an intensity varying at the surface according to a given law ; in particular the temperature may be given over one portion, while there is radiation over the remainder. These questions, which are not yet solved in their widest generality, have largely contributed to the direction taken by the theory of partial differential equations. They have called attention to types of determination of the integrals which would never have presented themselves if we had been restricted to a purely abstract point of view. Laplace’s equation has been already met with in hydrodynamics, and in the study of attraction varying inversely as the square of the distance. The latter theory brought to light elements of the most essential nature, such as the potential of single and double layers. Here we meet with analytical combinations of the highest importance, which have since been notably generalised. Green’s formula is a case in point. The fundamental problems of electrostatics are of the same order of ideas, and certainly the celebrated theorem on electrical phenomena in the interior of a hollow conductor, which Faraday rediscovered at a later stage by experimental means, knowing nothing whatever of Green’s memoir, was a notable triumph for theory. This magnificent aggregate has remained the type of the classical theories of mathematical physics, which seem to us to have almost attained perfection, and which have exercised, and still exercise, so happy an influence on the progress of pure analysis by suggesting to it the most beautiful problems. The theory of functions again will afford us a notable comparison. The analytical transformations brought into play are not distinct from those we have met with in the steady movement of heat. Certain fundamental problems in the theory of functions of a complex variable have lost their abstract enunciation and assumed a physical form, as in the case of the distribution of temperature on a closed surface of any connectivity whatever and without radiation, in thermal equilibrium, with two sources of heat which necessarily correspond to equal and opposite flows. Interpreting this, we find a question on Abelian integrals of the third species in the theory of algebraical curves.

Projective geometry derives both interest and importance from the fact that it assumes only the simplest and most fundamental axioms. The following sketch of the first principles, and proof of the fundamental theorem, is taken from Prof. Federigo Enriques' Projektive Geometrie (B. G. Teubner, Leipzig, 1903, 8vo, pp. xiv, 374). This work is specially adapted for school teaching. The first (Italian) edition appeared in 1898, and the axioms on which it is founded were explained, four years earlier in the Rendiconti R. 1st. Lombardo, Vol. 27, pp. 550-567.

Projective geometry is the geometry originated by von Staudt in the Geometrie der Zage, of which an excellent account by Prof. C A. Scott has already been given in the Mathematical Gazette, Vol. I., Nos. 19, 20, 22, pp. 307, 323, 363. Russell's very original Foundations of Geometry is the only English work which treats the subject from a strictly logical standpoint, but is unfortunately very unreliable as far as the mathematics is concerned.

Among those who have contributed to the development of Infinitesimal Geometry, Sophus Lie is distinguished by several important discoveries which place him in the first rank. He was not one of those who showed from childhood a very marked aptitude, and when he was leaving the University of Christiania in 1865, he was still hesitating between Philology and Mathematics. It was Plücker’s works which made him fully conscious for the first time of his vocation. He published, in 1869, his first paper on the interpretation of imaginaries in Geometry, and by 1970 he was in possession of the ideas which guided the whole of his career.

For the sake of those readers who have not seen the first part of this paper, and to whom the whole subject is new, I may state that the whole argument is based on the fact that any imaginary point such as (6 + i, 4 + 3i) has a simple vector x + iy = (6 + i) + i(4 + 3i) = 3 + 5i, and therefore may be vectorially plotted on to, or represented by, the real point [in this case (3, 5)] which has the same vector. This real point is thus the representative of all the imaginary points which have the same vector. The paper deals with the consequent relations between the representatives of conjugate imaginaries, especially when the imaginaries satisfy the equation of some given curve : in this case the two real points representing such a pair of conjugate imaginaries are called images of each other in the curve.

An article by Mr. G. H. Hardy in the March number of The Mathematical Gazette (coupled with certain other articles in recent numbers) raises several points which can hardly be allowed to pass without further discussion. It is proposed to discuss these points under three different heads, of which the first two are treated here while the third will find space in a later number of the Gazette.

The investigations of Professor Karl Pearson and his coadjutors on the applications of the theory of errors to statistics have greatly increased the importance and extent of the subject.

It is hoped that the following version of one of the methods of arriving at the so-called normal law of error may be of interest from an elementary point of view: affording, as it does, a simple application of the binomial theorem and some interesting graphical work.

The writer is not attempting to construct a new system of dynamics, but merely to present in a clear form what he believes to be the principles of our existing system, and he ventures to hope that such a presentation may be of some use to those teachers of dynamics who are not specialists, and perhaps to older students. The paper is not intended for experts.

It has appeared to him that in the various text-books written on this subject, these principles are at times forgotten, with the result that there are to be found many faults, some due to incompleteness, some to vagueness, some to reasoning in a “vicious circle,” and some to the attempt to make one equation give us two or even three unknowns.

In its revised form the Report of the M.A. Committee on the Teaching of Elementary Mechanics strikes me as being on the whole an exceedingly good and helpful one, though of course open to criticism in some of its details. On two or three points relating to Kinetics, I offer these remarks:

It is with the deepest sorrow that we record the tragic death of Ronald Hudson, at the early age of 28, while spending a holiday with a friend in Wales. He and his friend started from the Penygwryd Hotel early on Tuesday morning, September 20, and after climbing one of the Glydyrs descended to the foot of the Devil’s Kitchen by Lake Idwal, which they reached about one o’clock. Ronald Hudson was an enthusiastic rock climber, and would not be dissuaded from trying part of the ascent, although his companion was not sufficiently expert to go with him. A short half hour later he apparently dislodged a large rock, fell a considerable distance, and in all probability was instantly killed. When a search party with much difficulty found him some time after, he was dead. On the following Saturday he was laid to rest in Wandsworth Cemetery, by the side of his mother, whom he had lost when he was six years old. The heartfelt sympathy of his large circle of friends goes out to the sorrowing family, to his father, aunt and sisters, who have followed along the same path of knowledge as their brother with no less determination and success.

I Quite agree with Mr. Hardy (Gazette, vol. 3, p. 284) in his contention that the real basis of the power-series for sin x; and cos x is given by the two equations

But I am inclined to doubt whether at present the difficulties of exposition will not outweigh the advantages of his method (iii); and for this reason the following method may be worth considering, although it is a fairly obvious modification of my previous note (Gazette, vol. 3, p. 85). The only preliminary knowledge implied is that of the elementary differential calculus, and of Argand’s diagram.

One of the objects of a Congress such as this is to exhibit the connecting links of the different departments of Science, taking that term in its widest acceptation. Hence the organisers of this meeting have taken care that the relations between the different sections shall be exhibited in broad relief. In undertaking a sketch of this kind, which must necessarily be somewhat indefinite in character, we must forget that all is in all; as far as algebra and analysis are concerned, the follower of Pythagoras would be disconcerted by the extent of his task, remembering the celebrated formula of the School: “Things are numbers.” From this point of view my subject would be in-exhaustible. But I shall make no such pretensions, and for the best of reasons. By passing in rapid review the development of our science throughout the ages, and particularly in the last century, I hope to be able to give an adequate indication of the part played by mathematical analysis in its relations to other sciences.

The operation (c) of taking the complement of an angle (α) is of order (period) two, since 90° –(90° –α) = α. Similarly, the operation (s) of taking the supplement of an angle is of order two. If we perform these two operations in succession there results an operation (cs), which is of order four since it changes α into α + 90° Hence the group {c, s} generated by c and s is the dihedral rotation group of order 8. This fact follows directly from the general theorem, Two operations of order two generate the dihedral rotation group whose order is twice the order of their product.