George A. Gibson

On the History of the Fourier Series

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 Dirichlets 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.

Newton's Conception of a Limit as interpreted by Jurin and Robins respectively

In his recent book A History of the Conceptions of Limits and Fluxions in Great Britain from Neuton to Woodhouse (Chicago and London: The Open Court Publishing Company, 1919), as well as in a series of articles in the American Mathematical Monthly for 1915 on The History of Zenos Arguments on Motion , Mr Cajori discusses certain aspects of the conception of a limit, and treats in considerable detail the controversy between Jurin and Robins that arose out of the publication of Berkeleys Analyst . I gave an account of the controversy in a paper that appears in Volume XVII. of our Proceedings , and as Mr Cajoris estimate of the respective merits of the contributions by Jurin and Robins differs greatly from mine, and as the conception of a limit is fundamental in modern mathematics I venture to draw the attention of the Society to the matter.

The Theorem of Pythagoras

The parts of the figures are numbered to indicate a proof of the theorem by dissection. The congruence of parts on which the same number is marked can be demonstrated by geometry. The steps of the proof are in order (i), (ii), (iii). Sufficient importance is, perhaps, not attached to (i) and (ii) at this stage; for they give (1) the construction of a square equal to a given rectangle (2) the graphical construction of √2, √2 + 1.etc.

De la Vallée Poussin's Extension of Poisson's Integral

The integrals of §§ 5, 6, and 7 of the following paper were first established by C. de la Vallee Poussin in a memoir Sur quelques applications de lintegrale de Poisson ( Ann. de la Soc. sc. de Bruxelles , vol. 17, 1892–3). An analogous integral to that of § 5 was discovered by A. Hurwitz, who seems not to have been aware of de la Vallee Poussins memoir, and will be found under the title Sur quelques applications des series de Fourier in the Annales de lEcole normale , vol. 19, 1902. In view of the value of these integrals for the theory of the Fourier series, the discussion now given, which follows different lines from those of previous proofs, may be of some interest. The discussion turns chiefly on the Second Theorem of Mean Value which is quite as applicable to Poissons as to Dirichlets Integral.

A Reduction Formula for Indefinite Integrals

On p. 403 of Greenhills Calculus (2nd Ed.) the following sentence occurs:— “By differentiation of the integral with respect to A, B, or C we can deduce the results of For the evaluation of tlie typical form in which f ( x ) is a linear function, especially when A, B, etc., are given numbers, the method of differentiation does not seem very suitable; be that as it may, it may perhaps be of some interest to investigate a formula of reduction analogous to those in use for the integrals in which A x 2 + 2B x + C is replaced by a linear function and f ( x ) is a constant.

Some Properties of Parabolic Curves

If the tangent at a point P on the parabolic curve cy = x n meet the axis of x at M, it is a well-known property that the area between the radius vector OP and the are OP is n times that between the arc OP and the two tangents OM, MP, O being the origin and n > 1. The converse is also true; for taking any point O on a curve as origin and the tangent at O as axis of x , let us seek for the locus of P if the area between OP and the arc OP be n times the area between the arc OP and the tangents OM, MP.

A short notice of the additions to the Mathematical Theory of Heat since the transmission of Fourier's Memoir of 1811 to the French Academy

What is here printed contains merely a list of the memoirs and treatises that may perhaps be found useful for one who wishes to trace the progress of the mathematical theory of heat beyond the stage at which Fourier left it. As discussions of the Fourier series and integrals occur in almost every treatise on the Integral Calculus, I have omitted reference to these. Similar considerations have led me to omit references to the discussion of differential equations, except where these specially dealt with the problem of the conduction of heat.