8 REPORT — 1841. 



but not without an expression of the restrictions which the Academy put on 

 its favourable opinion. The committee appointed to examine and report on 

 the memoir, consisting of Laplace, Lagrange and Legendre, whilst they 

 agreed in proclaiming the novelty and importance of the subject, and in de- 

 claring that the equations are the true equations required by the conditions, 

 expressed a difficulty about the way in which they had been deduced, and 

 added, that the means employed to effect their integration left much to be 

 desired. Fourier never yielded to this judgment ; and accordingly he printed 

 his memoir exactly as it was written in the memoirs of the Academy for 

 1 825 and 1 826 : nor did he ever modify or extend his views, so far as we know. 

 He published his Treatise on Heat in 1822, which does not essentially differ 

 from his memoir. 



It is not necessary to trace all the circumstances which withheld from the 

 world these important investigations for so many years. We must not lay 

 all the blame on the Institute ; doubtless a part of it falls on Fourier himself. 

 The accounts which had appeared were scanty. They will be found in the 

 'Annales de Chimie,' iii. 250 (1816), iv. 128 (1817), vi. 259 (1817) ; ' Bulletin 

 des Sciences de la Societe Philomatique', 1818, p. 1, and 1820, p. 60 ; the 

 'Analyse des Trauvaux de 1' Academic,' 1820, Scc.hy Delambre. 



Whilst the memoir lay in the archives of the Institute, the labours of 

 Dulong and Petit had, by the establishment of another law, rendered it de- 

 sirable that the theorist should reconstruct his equations and extend his ana- 

 lysis. We can understand well enough why M. Fourier did not attempt 

 this. Whilst his own investigations lay unknown, he had no inducement to 

 extend or continue them : far less was he likely to take in hand a totally new 

 investigation which could hardly be expected to present results so beautiful 

 and symmetrical, and must, from their further approach to a correspondency 

 with the laws of nature, have withdrawn attention from his previous labours. 

 But we are astonished that M. Poisson, who laboured successfully in this as 

 in every other field of mathematical physics, did not see the necessity of 

 adopting axioms conformable to fact. We suspect that he and Lame, and 

 all the other writers who treated of the subject, were more anxious to pursue 

 a line of investigation which led to symmetric formulae, than one which 

 should lead to results conformable with the facts of experiment. 



The person who first attempted an investigation based on principles more 

 approaching to the probable law of nature was M. Libri. His memoir was 

 read to the Academy of Sciences in 1825, and is printed in the 'Memoires de 

 Mathematique et de Physique ' *, and in ' Crelle's Journal ' for 1831, vii. 1 16. 



The grounds of his investigations are, 1. That extra-radiation follows the 

 law of Dulong and Petit. 2. That conduction follows the law of Biot, La- 

 place and Fourier. He confines himself to the solution of one problem, as 

 the most simple that could be selected to illustrate his views. The problem 

 is the determination of the temperature of a ring heated 'at one point. The 

 author integrates the equation for the variable state of heat on the hypo- 

 thesis that the variation from the ordinary results which is introduced by 

 taking Dulong and Petit's law is a small quantity. Certain peculiarities in 

 his process of integration have drawn the attention of those who are inter- 

 ested in the subject to this memoir. The author of the present report was 

 the first to find fault with M. Libri's solution in 1837 f. Others have, since 

 that time, joined in the opinion ; amongst the rest M. Liouville. The paper of 

 this author, read to the Institute, and published in his 'Journal des Mathema- 

 tiques ' for 1838, has caused some little discussion on the subject, which the 

 reader will find in the ' Comptes Rendus ' for 1838, 39 and 40. 



* Florence, 1829. f Theory of Heat, p. 69. 



