26 
FIFTH REPORT- 
such as is given. The latter condition may again carry us among 
discontinuous functions. But the solution enables us to see 
that each concentric spherical shell of which the globe is com¬ 
posed will diminish in temperature in a geometrical progression 
with respect to the time, as above stated. Also it appears that 
after the lapse of a considerable time of undisturbed cooling, the 
distribution of temperature in the sphere decreases in proceeding 
from the centre to the surface, as the quotient of the sine of an 
arc divided by the arc, the lengths of the arcs being the distances 
from the centre, and the radius, to which such lengths are made 
arcs, depending on the conductive and radiative powers. 
Besides these solutions of problems respecting the distribution 
of heat, Fourier has given solutions in other forms, involving 
definite integrals. By means of such integrals, discontinuous 
functions may be expressed; and the functions, of which these 
definite integrals are taken, involve the given function repre¬ 
senting the original distribution of the heat. For instance, the 
problem of the propagation of heat in an infinite line is ex¬ 
pressed by an integral of this kind, given by Laplace as the solu¬ 
tion of an equation of partial differences*. 
By such analytical artifices Fourier solved a number of the 
problems belonging to this subject; purposely varying his me¬ 
thods, as he says, 44 a fin de multiplier les moyens de solution dans 
une matiere aussi nouvellet.” Thus, besides the cases already 
mentioned, of a rectangular solid and a solid sphere, he treats of 
an u ar mi lie,” or ring, the properties of which are somewhat cu¬ 
rious in reference to this subject; also of a solid cylinder, of a 
rectangular prism, and of a cubej. A succeeding part of the 
work contains the laws of the propagation of heat in an infinite 
solid. This case is of importance, in as much as the conclusions 
are applied to the propagation of heat in the mass of the earth; 
which, for such purposes, may be considered as of infinite dimen¬ 
sions. These conclusions deserve to be stated.—When any 
part of a solid mass is affected by alternations of greater or less 
temperature, (as the surface of the earth is affected by diurnal 
and annual alternations,) the following are the results. 
1st. The range of the oscillations of temperature becomes 
smaller as we recede from their origin, and at last they become 
* u — fd qe q <p(x 2 q kt), the solution of — = 1c < LA 
f Theorie de la Chaleur, p. 452. 
X “ M. Lame, professeur de physique a l’Ecole Polytechnique, has determined 
the law of temperature of all the points of a homogenous ellipsoid brought to a 
permanent condition. The expression of this law depends on elliptical func¬ 
tions.’—Poisson, Theorie de la Chaleur , p. 4. 
