150 



MATHEMATICAL AND PHYSICAL SCIENCE. 



[Diss. VI. 



yet never acquire that temperature, even approxi- 

 mately ; and physically, considering that radiation 

 proceeds not from a mathematical surface, but from 

 a material physical stratum of an imaginable thick- 

 ness, but which rapidly absorbs the emanations pro- 

 ceeding from the inferior particles, he proved that 

 the attenuation due to oblique emanation will fol- 

 low Leslie's law, independent of the precise rate of 

 absorption in traversing the physical surface. 

 (668.) Fourier takes extraordinary pains to define and 

 Definition j us tify every step of his demonstrations. He has the 

 ing power, g rea * merit of having first given a clear definition of 

 conducting power, or conductivity proper, which is 

 this : " The number of units of heat (measured by 

 the weight of ice which it can melt) passing in unit 

 of time across a square unit of surface of an infinitely 

 extended plate bounded by two parallel surfaces at 

 unit of distance which are respectively maintained 

 at the freezing and boiling temperatures (unit of dif- 

 ference of temperature)." 1 In like manner, the " ex- 

 and of terior conductivity" expresses the number of units 

 " exterior of heat parted with by unit of surface to the air and 

 Yit ctl " surrounding space, when the difference of their tem- 

 perature amounts to unity. 



(669.) Account of the Theorie Analytique de la Chaleur. 

 ouner's rp^e p ro bi ems considered by Fourier in his Theorie 



Theorie An- . f ..', 



alytique de Analytique^ reter principally to the propagation oi 

 laChaleur. heat in homogeneous conducting solids of definite 



forms, and in some cases maintained in certain parts 



at fixed temperatures. 

 (670.) The number of examples fully worked out is very 



ofp?owSns Sma11 ' but the y ma y be referred to tne following 

 solved by classes : (1 .) When some part of a solid has an inde- 

 Mm. finite source of heat applied to it, the remaining sur- 

 face being exposed to the air, or having determinate 

 temperatures maintained at certain parts. In this 

 case, the state of the solid in regard to heat is per- 

 manent, or independent of time ; and the problem is to 

 assign the temperature of each part, and the flow of 

 heat through that part in a given direction. (2.) To 

 assign the temperature of every point of a solid pri- 

 mitively heated, either uniformly or after any assigned 

 law, and at any given moment. (3.) To solve the 

 last question only in the case where the cooling at the 

 surface has been going on for an exceedingly long 

 time. 



(671.) of t ne fi rs t class of problems, the slender bar heated 

 conditio^ b y a consent source of heat at one end, and exposed 

 of heat in a to the cooling influence of radiation and of the air, 

 slender which had been treated of by Lambert, is the simplest 

 and most important. The temperature of any thin 



bar. 



slice perpendicular to the axis of the bar, is the 

 result on the one hand of the heat which it acquires 

 from the hotter slice nearest to it on the side of the 

 source of heat ; and on the other, of the heat with 

 which it parts to the slice next beyond and also to 

 the air in contact with the exterior surface of the 

 slice and by radiation from the same surface. The 

 solution of this problem is that of a simple differential 

 equation of the second order, and the result is the 

 diminishing geometrical progression of temperature 

 already mentioned. This has been approximately con- 

 firmed by some careful experiments of M. Biot, which 

 indeed are nearly the best which we yet possess on the 

 subject. But instead of drawing from them, as he 

 does, an argument for the accuracy of the Newtonian 

 law of cooling, the diminution of temperature along 

 the bar is far more rapid at first, and less afterwards 

 than that law indicates. In fact, the apparent agree- 

 ment of the formula is owing to the use, in a case to 

 which it does not correctly apply, of that often mis- 

 applied rule of the doctrine of chances the method 

 of least squares. 



The solution of another case of stationary temper- (672.) 

 ature, an indefinite solid bounded* by three infinite In an infi( 

 planes (two of which, B, C, are parallel, and the third, nite so1 " 

 A, perpendicular to both) having determinate tem- 

 peratures, requires the introduction of a species of 

 analysis, in which Fourier acquired great dexterity, 

 but which is of so subtle a kind as to have created 

 doubts in the minds of the committee of the Institute 

 to which the Memoir was referred, and to have been 

 a source of some controversy and much discussion 

 since. Fourier contrives to express, by an infinite 

 trigonometrical series, the law of temperature in such 

 a solid, which shall not only satisfy the differential 

 equation of the equilibrium of heat, but also the 

 conditions of temperature at the bounding planes, B 

 and C, which being zero by the problem, the value 

 of the temperature which, up to that point was finite, 

 suddenly comes to nothing, and has no value beyond. 

 This problem leads to a long digression on the pos- 

 sibility of expressing by trigonometrical series, quan- 

 tities which vary according to any conceivable law 

 and of determining the co-efficients of the successive 

 powers of the sines and cosines employed. The 

 theorem to which Fourier is led, in which any function 

 of a; is expressed by a series of definite integrals, in- 

 cluding sin x and cos , is known by his name. 



The problem, however, which Fourier most elabo- (673.) 

 rately treated, belongs to the 2d and 3d class, Movement 

 namely, the cooling of a sphere primitively heated of ^ ei 



sphere. 



1 Let F be the flux of heat measured as above, K the constant of interior conductivity, z an ordinate measured across the 

 thickness of the plate, and V he temperature of the stratum of which Z is the ordinate ; then F= K ; in the permanent 

 state the temperature varies uniformly from stratum to stratum. When the thickness and difference of temperature both are 

 equal to unity, K=F. The expression F= K manifestly expresses the Newtonian Law, interpreted by Fourier as 

 stated above. 



