6 REPORT — 1841. 



elastic fluid, considerable difficulties present themselves. We do not know 

 that the difficulties are real ; we think, as Mr. Whewell hints*, that they are 

 introduced by an arbitrary assumption concerning infinitesimal magnitudes. 

 One difficulty is as follows : If heat flow from one point or place to another, 

 the variation of temperature is a quantity of the first order ; whilst if we ob- 

 tain the variation by estimating the gain and loss of heat which that point or 

 plane receives, we shall find it to be a quantity of the second order. Biot, 

 who in 1804 read to the Institute a short memoir on this subject fj was con- 

 strained to leave his fundamental equation without demonstration on this 

 account {. The difficulty is supposed to have been removed by Laplace §, 

 who does indeed present reasoning bearing with some weight on the subject. 

 But we could have wished that he had distinctly answered the following 

 question. If three equal, small, contiguous slices of a bar be conceived col- 

 lected each at its middle plane, will the quantity of heat which in a given time 

 passes from the first to the second, or from the second to the third, depend 

 on the (small) thickness of the slices or not ? Fourier doubtless saw that it 

 would not, and therefore, instead of reasoning on the difference of heat be- 

 tween two portions of the body directly, he fixes his attention on the Jlow of 

 heat across a plane. His reasoning is as follows : — A homogeneous body is 

 supposed to be traversed by two parallel planes whose distance is e, of the 

 lower of which every point has the same temperature a, and of the upper a 

 different and less temperature h. Then, if v represent the temperature at 

 any intermediate point at the distance z from the lower plane, the expression 



a — h 



V ■= a — '■^— z being supposed to be once established as the law of the 



temperature at all points, no change will take place in the state of heat of the 

 body II . To prove this he takes two points at equal distances from the plane 

 whose temperature is v, the one above, the other below it, and shows that the 

 excess of the temperature of the lower above v is exactly equal to the defect 

 of temperature of the upper from v. He then concludes thus : " It follows 

 that the quantity of heat transmitted by the lower point to the middle one is 

 the same as that which the middle one transmits to the upper, for all the 

 elements loldch concur to determine this quantity of transmitted heat are the 

 samef^." Thus M. Fourier's hypothesis of conduction is, that the flow of heat 

 depends on the difference of temperature ; or as he gives it, " the flow of heat 

 across a given plane, whose distance from some fixed plane is a and tem- 

 perature V, is proportional to - — " This we regard as the first law of con- 

 duction. 



No doubt M. Fourier has confounded heat with temperature ; but this 

 confusion is merely a confusion of terms ; the reasonings and results are un- 

 affected by it. 



M. Poisson, founding his theory on molecular interchange, and having in 

 view Dulong and Petit's law of radiation, admits another law of conduction. 

 This law is thus expressed : " The change of heat between two points depends 

 on the product of the difference of temperatures of those points, and of a 

 function of their positions and temperatures **." In M. Poisson's earlier in- 



* History of the Inductive Sciences, vol. ii. p. 470. 

 t It is printed in the BibUotheque Britannique. 

 X See Biot, Traite de Physique, torn. iv. p. 669. 



§ Laplace, Memoires de rAcademie, 1809, p. 332. Connaissance des Tems, 1823, and 

 Mecanique Celeste, liv. ii. || Fourier, Theorie de la Chaleur, p. 47. 



H Ibid, p. 49. ** Poisson, Theorie de la Chaleur, art. 48. 



