In a paper “On the Theory of Long Waves” recently published, I drew attention to a point of some interest in the theory of the solution of partial differential equations by what is usually termed the method of reciprocation. As the subject was apart from the main object of the paper, however, I there noticed the peculiarity but briefly, and so I have thought that it might be of some use to enter into the matter here somewhat more fully. On first adverting to the point, it was my opinion that it could hardly have escaped the notice of those who had made much application of the method of solution by reciprocation, though I had not myself seen any reference made to it; but as I have not had my attention drawn to any such notice since the publication of the paper last March, it seems probable that it has not previously been discussed.

The second law of motion may be expressed as a dimensional equation in the form

where the meanings of the quantities are obvious.

If we cut out the factor m from each side, we may write this in the usual form

The general solution is

where

We propose to obtain certain results involving areal and trilinear co-ordinates, by a uniform method of changing to Cartesian co-ordinates with two sides of the triangle of reference as axes.

The earliest author to whom the discovery of the nine-pointcircle has been attributed is Euler, but no one has ever given a reference to any passage in Euler's writings where the characteristic property of this circle is either stated or implied. The attribution to Euler is simply a mistake, and the origin of the mistake may, I think, be explained. It is worth while doing this, in order that subsequent investigators may be spared the labour and chagrin of a fruitless search through Euler's numerous writings.

If in a triangle ABC, points are taken on the sides such that

then the radical axis of the circles PQR, P′Q′R′ passes through the centroid and “S.” points of ABC; and if QR, Q′R′ cut in 1, RP, R′P′ in 2, PQ, P′Q′ in 3, then the equation to the circle 123 is

It can be shown as follows that Newton's Theorem can be derived from elementary considerations without making use of the idea of an equation and its roots.

Of late years there has arisen a clique of vector analysts who refuse to admit the quaternion to the glorious company of vectors. There are others again who take exception to some of Hamilton's most fundamental principles, and make corrections as they deem them, which logically revolutionise the whole basis of the calculus.

For a proof of the following theorem due to Frégier, see Salmon's Conic Sections, p. 175, or Gergonne's Annales, VI. 231 (1816).

It is well known (see Thomson and Tait, §§ 517, 518) that a spherical shell, whose surface-density is inversely as the cube of the distance from an external point, as well as a solid sphere whose density is inversely as the fifth power of the distance from an external point, are centrobaric. The centre of gravity is, in each case, the “image“ of the external point.

When a student of mathematics commences the study of a subject which involves the assimilation of what are, to him, fundamentally new ideas, his progress is, as a rule, slow at first. And, even after he has become accustomed to these ideas, he may still require a long course of laborious practice, before he can attain to that mastery of the method which would enable him to use it as a powerful aid to research. Thus students, familiar with geometrical methods, when first commencing the study of Cartesian analysis, require much practice before they can call up mentally the geometrical figure corresponding to a given equation. And, the more general the new method is, the greater is the difficulty felt to be. So, in Hamilton's system of quaternions, the difficulty of assimilation is greater than it is in the Cartesian analysis. And it seems as if it were for this reason that, in recent years, attempts have been made, by men of known mathematical ability, to smooth the paths.

In 1873, at the Lyons meeting of the French Association for the Advancement of the Sciences, Monsieur Emile Lemoine called attention to a particular point within a plane triangle which he called the centre of antiparallel medians. Since that time the properties of this remarkable point and of the lines and circles connected with it have been investigated by various writers, foremost among whom is Monsieur Lemoine himself. The results obtained by them are so numerous (indeed every month adds to their number) and so widely scattered through the mathematical periodicals of the world that it would be a task of considerable magnitude to make even an undigested collection of them. It is the purpose of the present paper to state those properties of the point which had been discovered previously to 1873. A short sketch of some of them will be found at the end of a memoir read by Monsieur Lemoine at the Grenoble meeting (1885) of the French Association, and in a memoir by Monsieur Emile Vigarié at the Paris meeting (1889) of the same Association. The references given by Dr Emmerich in his Die Brocardschen Gebilde (1891) are very valuable. It is a pity they are not more explicit.

In this paper the laws of addition and subtraction of vectors were considered, and examples of their extreme usefulness in geometrical applications were given.

In triangle ABC, AD, BE, CF are concurrent at O; through O parallels are drawn to EF, FD, DE, meeting the sides of ABC in L, M, P, Q, S, T, and the sides of DEF in L′, M′, P′, Q′, S′, T′. The two hexagons LMPQST, L′M′P′Q′S′T′ thus formed have the following properties:

(1) The sides L′M′, P′Q′, S′T′ of the latter are parallel to the sides of ABC.

The mutual action of two electrified bodies was regarded by Maxwell as transmitted by a medium. According to him the stress in the medium consists of a “tension like a rope” along the lines of electrical force whose intensity per unit of area is R2/8π, where R is the resultant electric intensity, and of a pressure numerically equal to this in all orthogonal directions. Maxwell's remarks are somewhat vague but his notation is strongly suggestive of an elastic solid medium. It has, however, been pointed out by Minchin that Maxwell's stress system would not in an ordinary elastic solid give origin to strains consistent with the “equations of compatibility” which the theory of elastic solids supplies. Considerable interest still attaches to the theory of an elastic solid medium propagating stresses equivalent to the action between distant bodies of forces varying inversely as the square of the distance. For in the first place, it has been pointed out that the stress system given by Maxwell does not constitute a unique solution of his equations; and, in the second place, it has been suggested that some medium must exist for the transmission of gravitational forces. The statical problem of the propagation of gravitational forces by an isotropic elastic medium has been treated by Minchin. His treatment how ever neglects a certain surface condition. I have thus thought it worth while to consider the problem independently, employing the ordinary surface conditions. The first part of the paper is devoted more especially to the electrostatic problem, but the elastic solid problem is essentially the same throughout.

Three main aws regulate the treatment of ordinary algebraic quantities. These are the Associative Law, the Distributive Law, and the Commutative Law. If a, b, c, … , represent quantities dealt with in the algebra, the associative law of multiplication asserts that a(bc)=(ab)c, where the brackets have the usual meaning that the quantity within them is to be regarded as a single quantity: the distributive law of multiplication asserts that (a + b)(c + d)=ac + bc + ad + bd: and the commutative law gives ab = ba. With regard to addition, the associative law asserts that (a + b) + c = a +(b + c): and the commutative law gives a + b = b + a.

The theorem, “If upon the sides of a triangle as diagonals parallelograms be described, whose sides are parallel to two given lines, then the other three diagonals will intersect in the same point,“ occurs in Hutton's Course of Mathematics, 12th ed., vol. II., p. 191.

The treatment of the Fourier Series, that is, of the series which proceeds according to sines and cosines of multiples of the variable, is in most English text-books very unsatisfactory; in many cases it shows almost no advance upon that of Poisson and, even where a more or less accurate reproduction of Dirichlet's investigations is given, there is no attempt at indicating the advantages it possesses over the so-called proof of Poisson. Nor is the uniformity of the convergence of the series so much as mentioned, not to say discussed. I have therefore thought it might be useful to give a fairly complete outline of the historical development of the series so far as the materials at my disposal allow. I do not think that any important contribution to the theory is omitted, but, as I indicate at one or two places, there are some memoirs to which I have not had access and which I only know at second hand.