SECT. VII.] LITERATURE. 209 



quantites periodiqucs. Or, more briefly, in his TraiU de Mecanique, Arts. 325328. 

 Poisson s original memoirs on the subject were published in the Journal de VEcole 

 Poll/technique, Cahier 18, pp. 417 489, year 1820, and Cahier 19, pp. 404509, 

 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. VIH. pp. 533 556. 

 On the Critical i-alucs of the sums of Periodic Series. Section I. Mode of ascertain 

 ing 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. I. Arts. 7577. 

 Donkin, Acoustics, Oxford, 1870, Arts. 72 79, and Appendix to Chap. rv. 

 Matthieu, Cours de Physique Mathematique, 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 con 

 vergence des series trigonome triques qui servent a rcpresenter une fonction arbitraire 

 entre les limites donnees. The methods of this memoir thoroughly deserve at 

 tentive 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 Phyaik, 

 Berlin, 1837, Band i. pp. 152 174. Ueber. die Darstellung ganz willkuhrlicher 

 Functioncn 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 imd Cosinussen der Vielfachen eines Winkel* fortachreit en- 

 den Eeihe. 



Bessel, Astronomische Nachrichten, Altona, 1839, pp. 230 238. Ueber den 



Amdruck einer Function $ (x) durch Cosinusse und Sinusse der Vielfachen von x. 



The writings of the last three authors are criticised by Biemann, Gesammelte 



Mathematische Werke, Leipzig, 1876, pp. 221 225. Ueber die Darstellbarkeit einer 



Function durch eine Trigonometrische Eeihe. 



On Fluctuating Functions and their properties, a memoir was published by 

 Sir W. K. Hamilton, Transactions of the Royal Irish Academy, 1843, Vol. 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 BAG, whose end A 

 is constantly heated, whilst its two infinite edges B and C are 

 maintained at the temperature 0. 

 F. H. 



