CHAP. VI., 6.] 



HEAT. FOURIER. 



151 



in a given way, but so that points equi-distant from 

 the centre have a common temperature. 1 It might 

 be expected that the symmetry of the conditions would 

 admit of a simple solution ; and such indeed was 

 sought by Professor Playfair long before, in a paper 

 in the Edinburgh Transactions, with reference to Geo- 

 logical Speculations. It was, however, by no means 

 sufficiently general. It is shown by Fourier, that 

 even in the simplest supposable case that where the 

 temperature of the sphere was originally uniform 

 the resulting expression for the temperature of any 

 stratum at any time, though capable of algebraic ex- 

 pression, cannot be assigned in finite terms, and no 

 attempt has been made to evaluate it generally. The 

 special cases which have been considered, are when 

 the sphere is extremely small, or has been cooling 

 for a very long time. 



(674.) The extreme complication of even such apparently 

 lf simple cases when solved in all their generality con- 

 in in the s ' s * s * n ^is, that *^ e n ' ow f h ea t across each part is 

 oblem of in fact dependent at each instant on the state of heat 

 3 Sphere. j n g^^ o ther part, and this distribution of the whole 

 is equally unknown with the local distribution on 

 which the movement of heat in each part depends. 

 To this must be added the peculiarity of conditions 

 at the surface, where the temperature undergoes an 

 abrupt change. The law according to which the su- 

 perficial particles radiate heat, is also different from 

 that according to which they receive it from the in- 

 terior. When a body uniformly heated to a consider- 

 able temperature cools in air, the subtraction of heat 

 from the surface commences with great rapidity, the 

 excess of temperature of the superficial particles being 

 very much greater than it ever can be afterwards. 

 The drain of heat from the interior of the sphere is 

 at first nearly imperceptible. The exterior cooling 

 will by and by become slower; and the degree in 

 which it takes place, will depend upon the rapidity 

 with which the conducting mass of the sphere is able 

 to supply fresh heat to the surface ; then the rate of 

 superficial cooling will become relatively somewhat 

 accelerated, and a fresh drain will take place from 

 the interior towards the surface. It is only in the 

 case of exceedingly small bodies, or those of an in- 

 finite degree of conductivity, that the body will cool 

 according to a simple law. In all other cases, there 

 will be characteristic periodical inflections in the 

 course of cooling. These, no doubt, are represented 

 in Fourier's series, if it could be numerically calcu- 

 lated ; and it is to be desired that some attempt were 

 made to represent it approximately. When the cool- 

 ing has endured for a very long time, these gushes 

 of heat cease altogether; the surface has attained 

 nearly to the temperature of the surrounding space, 

 and the drain of heat from the interior is so slow, 



that the progression of temperature is very gradually 

 and slowly disturbed. 



Application to the Thermal condition of the Earth. (676.) 

 This last case has been discussed by Fourier with Applica- 

 great address, relatively to the present condition of*" 

 the Earth, considered as a mass which has been once c ipi es to 

 at a high temperature, of which we have evidence the ther- 

 in the general increase of heat as we descend in mines, al condi 

 or when we penetrate its crust by Artesian wells, garth * 

 The surface of our planet receives a large amount of 

 heat annually by absorbing the sun's rays, but parts 

 with it by radiation into free space, to an extent which 

 preserves a sensible uniformity of temperature from 

 age to age. The considerations connected with the 

 subject are these : (1.) The proper heat of the earth, 

 and how much heat reaches the surface from the 

 (perhaps) still incandescent interior ; (2.) How much 

 heat do we receive from the sun, what share of it 

 enters the surface, and how far, and according to 

 what periods does the influence of the seasons extend 

 below the surface ? (3.) What is the amount of refri- 

 geration of the earth's surface ? how does the atmo- 

 sphere affect it ? and, if the cooling be due entirely 

 to radiation, what are we to set down for the temper- 

 ature of space, so as to account for the heat lost ? 



First, As to the proper heat of the globe. Not to (676.) 

 go further back than the last century, the incandes- ?" he P^P ei 

 cence of the earth's nucleus was assumed as very pro- globe . 

 bable by Buffon and other popular writers, and their 

 opinions were, on the whole, confirmed by the pro- 

 gress of observation. The existence of volcanoes was, 

 of course, an obvious argument ; another was, that 

 if the earth were once hot it must be still cooling ; con- 

 sequently, climates are continually becoming colder, 

 especially in the Arctic regions, which it was supposed 

 depended most on the supply of heat from within. In 

 evidence of this change were quoted, not only the 

 remains of elephants found interred with flesh and 

 skin in the midst of Siberian ice, but also the un- 

 equivocal proofs, so well known to geologists, that 

 at a period indefinitely more remote, tropical plants 

 of gigantic growth, and animals of a class which now 

 frequent only southern seas, appear to have lived 

 and flourished in high northern latitudes. But even 

 in the time of Buffon, attention was directed to a fact 

 yet more important for the theory of heat, namely, 

 the increase of temperature observed in deep mines. 

 Notwithstanding many sources of doubt and confu- manifested 



sion, such as the heat from candles, and from men at b y tbe in ' 



' . . . crease of 



work in mines, from chemical changes in some cases, jj eat i n 



and from the increased density of the air, the fact of mines, 

 the increase is now well established, and also its 

 rate of progression with more certainty than in the 

 time 6f Fourier. On an average (including the best 

 data of all, those yielded by Artesian springs), the 



1 The problem of the Armil or ring, heated at one or several points, is one which offers considerable facilities for its solution, 

 but the results are of little utility, the form being so peculiar ; and even as a test of theory, the variations of temperature from 

 point to point are insufficient under the limited conditions in which the numerical solution is practicable. 



