948 
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, 
Fourier takes extraordinary pains to define and 
Definition justify every step of his demonstrations. He has the 
of conduct- ‘ ° ° eas 
ing power, great 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).”! 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 
conducti- surrounding space, when the difference of their tem- 
(668.) 
ity.” 3 
says perature amounts to unity. 
(669.) Account of the Théorie Analytique de la Chaleur. 
ho —tThe problems considered by Fourier in his Théorie 
alytique de Analytique, refer principally to the propagation of 
laChaleur. heat in homogeneous conducting solids of definite 
forms, and in some cases maintained in certain parts 
at fixed temperatures. 
(870.) The number of examples fully worked out is very 
erst small, but they may be referred to the following 
problems < . 
solved by Classes :—(1.) When some part of a solid has an inde- 
him. 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. 
: AS fe wy , OF the first class of problems, the slender bar heated 
conditt yal by a constant source of heat at one end, and exposed 
of heat inato the cooling influence of radiation and of the air, 
MATHEMATICAL AND PHYSICAL SCIENCE. 
{Diss. VI. 
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 ease 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, ite soled, 
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 w is expressed by a series of definite integrals, in- 
cluding sin # and cos , is known by his name. 
The problem, however, which Fourier most elabo- 673.) 
sleaioe which had been treated of by Lambert, is the simplest rately treated, belongs to the 2d and 3d class,— Movement 
“ and most important. The temperature of any thin namely, the cooling of a sphere primitively heated ° heat ina 
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 2 is the ordinate; then F= oe od 4 
ro 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= -«? manifestly expresses the Newtonian Law, interpreted by Fourier as 
stated above, 
! 
