1st. The series arranged according to sines or cosines of multiple arcs are always convergent; that is to say, on giving to the variable any value whatever that is not imaginary, the sum of the terms converges more and more to a single fixed limit, which is the value of the developed function. 2nd. If we have the expression of a function f(x) which corresponds to a given series a+b cos x + c cos 2x + d cos 3x + e cos 4x + &c., and that of another function p (x), whose given development is a+ẞ cos x + y cos 2x + 8 cos 3x + e cos 4x + &c., it is easy to find in real terms the sum of the compound series aa+bB+cy + dd+ee + &c.,1 and more generally that of the series az + bẞ cos x + cy cos 2x + do cos 3x + ee cos 4x + &c., which is formed by comparing term by term the two given series. This remark applies to any number of series. 3rd. The series (P) (Art. 234) which gives the development of a function F(x) in a series of sines and cosines of multiple arcs, may be arranged under the form + cos x F(a) cos ada + cos 2x | F (a) cos 2ad1+ &c. + sin x ( F (a) sin ada + sin 2x ( F (a) sin 2ad1+ &c. a being a new variable which disappears after the integrations. We have then * F (x) = [] F (a) da {}} + cos x cos a + cos 2x cos 2x + cos 3x cos 3a + &c. &c. }, + sin x sin a + sin 2x sin 2x + sin 3x sin 3a + &c. 1 We shall have 0 ᅲ (2) ¢ (2) dx=aan + }n {B+c+.... [R. L. E.] Hence, denoting the sum of the preceding series by 1 2 2 − a)}. The expression + cos i (x − a) represents a function of x and a, such that if it be multiplied by any function whatever F(a), and integrated with respect to a between the limits aπ and a=π, the proposed function F (2) becomes changed into a like function of a multiplied by the semi-circumference π. It will be seen in the sequel what is the nature of the quantities, such as 1 2 +Σcos i (x − a), which enjoy the property we have just enunciated. 4th. If in the equations (M), (N), and (P) (Art 234), which on being divided by r give the development of a function f(x), we suppose the interval r to become infinitely large, each term of the series is an infinitely small element of an integral; the sum of the series is then represented by a definite integral. When the bodies have determinate dimensions, the arbitrary functions which represent the initial temperatures, and which enter into the integrals of the partial differential equations, ought to be developed in series analogous to those of the equations (M), (N), (P); but these functions take the form of definite integrals, when the dimensions of the bodies are not determinate, as will be explained in the course of this work, in treating of the free diffusion of heat (Chapter IX.). Note on Section VI. On the subject of the development of a function whose values are arbitrarily assigned between certain limits, in series of sines and cosines of multiple arcs, and on questions connected with the values of such series at the limits, on the convergency of the series, and on the discontinuity of their values, the principal authorities are Poisson, Théorie mathématique de la Chaleur, Paris, 1835, Chap. vII. Arts. 92-102, Sur la manière d'exprimer les fonctions arbitraires par des séries de quantités périodiques. Or, more briefly, in his Traité de Mécanique, Arts. 325–328. Poisson's original memoirs on the subject were published in the Journal de l'École Polytechnique, Cahier 18, pp. 417–489, year 1820, and Cahier 19, pp. 404—509, year 1823. De Morgan, Differential and Integral Calculus. London, 1842, pp. 609–617. The proofs of the developments appear to be original. In the verification of the developments the author follows Poisson's methods. Stokes, Cambridge Philosophical Transactions, 1847, Vol. vi. pp. 533–556. On the Critical values of the sums of Periodic Series. Section I. Mode of ascertaining the nature of the discontinuity of a function which is expanded in a series of sines or cosines, and of obtaining the developments of the derived functions. Graphically illustrated. Thomson and Tait, Natural Philosophy, Oxford, 1867, Vol. 1. Arts. 75-77. Donkin, Acoustics, Oxford, 1870, Arts. 72-79, and Appendix to Chap. IV. Matthieu, Cours de Physique Mathématique, Paris, 1873, pp. 33-36. Entirely different methods of discussion, not involving the introduction of arbitrary multipliers to the successive terms of the series were originated by Dirichlet, Crelle's Journal, Berlin, 1829, Band iv. pp. 157–169. Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre les limites données. The methods of this memoir thoroughly deserve attentive study, but are not yet to be found in English text-books. Another memoir, of greater length, by the same author appeared in Dove's Repertorium der Physik, Berlin, 1837, Band 1. pp. 152–174. Ueber die Darstellung ganz willkührlicher Functionen durch Sinus- und Cosinusreihen. Von G. Lejeune Dirichlet. Other methods are given by Dirksen, Crelle's Journal, 1829, Band iv. pp. 170-178. Ueber die Convergenz einer nach den Sinussen und Cosinussen der Vielfachen eines Winkels fortschreitenden Reihe. Bessel, Astronomische Nachrichten, Altona, 1839, pp. 230-238. Ueber den Ausdruck einer Function (x) durch Cosinusse und Sinusse der Vielfachen von x. The writings of the last three authors are criticised by Riemann, Gesammelte Mathematische Werke, Leipzig, 1876, pp. 221-225. Ueber die Darstellbarkeit einer Function durch eine Trigonometrische Reihe. On Fluctuating Functions and their properties, a memoir was published by Sir W. R. Hamilton, Transactions of the Royal Irish Academy, 1843, Vel. xix. pp. 264-321. The introductory and concluding remarks may at this stage be studied. The writings of Deflers, Boole, and others, on the subject of the expansion of an arbitrary function by means of a double integral (Fourier's Theorem) will be alluded to in the notes on Chap. IX. Arts. 361, 362. [A. F.] SECTION VII. Application to the actual problem. 236. We can now solve in a general manner the problem of the propagation of heat in a rectangular plate BAC, whose end A is constantly heated, whilst its two infinite edges B and C are maintained at the temperature 0. F. H. 14 Suppose the initial temperature at all points of the slab BAC to be nothing, but that the temperature at each point m of the edge A is preserved by some external cause, and that its fixed value is a function f(x) of the distance of the point m from the end 0 of the edge A whose whole length is 2r; let v be the constant temperature of the point m whose co-ordinates are x and y, it is required to determine v as a function of x and y. -my The value vae sin mx satisfies the equation d'v d'v + =0; dx dy a and m being any quantities whatever. If we take m=i‡, i being an integer, the value ae - is πα r sin vanishes, when x=r, whatever the value of y may be. We shall therefore assume, as a more general value of v, If y be supposed nothing, the value of v will by hypothesis be equal to the known function f(x). We then have Σπα πα f(x)= a, sin + a sin +a, sin + &c. The coefficients a,, a, a,, &c. can be determined by means of equation (M), and on substituting them in the value of v we have 237. Assuming r = π in the preceding equation, we have the solution under a more simple form, namely 1 Tv = e* sin æ ƒ ƒ (x) sin ædæ +eTM sin 2x [ƒ (x) sin 2ædæ + e3 sin 3x [ƒ (2) sin 3ædæ + &........ (a), or 1 πV= ["ƒ (2) da (e" sin x sin a + eTM sin 2æ sin 2x + sin 3x sin 3x + &c.) e a is a new variable, which disappears after integration. If the sum of the series be determined, and if it be substituted in the last equation, we have the value of v in a finite form. The double of the series is equal to e[cos (x − a) — cos (x + a)] + e ̄2 [cos 2 (x − a) — cos 2 (x+a)] denoting by F (y,p) the sum of the infinite series πυ Ξ cos (x+a) - e 2 (e" - e) sin x sin a [e” − 2 cos (x − a) + e ̄"] [eo − 2 cos (x+a) + e |