M. POISSON ON THE MATHEMATICAL THEORY OF HEAT. 133 
general case by a quadruple integral, which can always be reduced to 
a double integral like each of the other parts. By the method which 
I have used to effect this reduction we obtain the value of different de- 
finite integrals, which it would be difficult in general to determine in a 
different manner, and the accuracy of which is verified whenever they 
enter into known formule. 
CuapTer XI. On the Distribution of Heat in certain. Bodies, and 
especially in a homogeneous Sphere primitively heated in any Manner.— 
It is explained how, in every case, the complete expression of exterior 
temperature, which may depend on the different sources of heat, and 
which must be employed in the equation of the motion of heat relative 
to the surface of bodies submitted to their influence, will be formed. 
After having enumerated the different forms of bodies for which we 
have hitherto arrived at the solution of the problem of the distribution 
of heat, the complete solution is given for the case of a homogeneous 
rectangular parallelopiped the six faces of which radiate unequally. 
In order to apply the general equations of the fourth and fifth chap- 
ters to the case of a homogeneous sphere primitively heated in any 
manner, the orthogonal coordinates in them are transformed into polar 
coordinates; the temperature at any instant and in any point is then 
expressed by means of the general series of Chapter VIII., and of the 
integrals found in Chapter VI.; the coefficients of that series are next 
determined according to the initial state of the sphere, by supposing at 
first the exterior temperature to be zero: by the process already em- 
ployed in the preceding Chapter, this solution is finally extended to 
the case of an exterior temperature, varying with the time and from 
one point to another. Among the consequences of this general solu- 
tion of the problem the most important is that for which we are in- 
debted to Laplace ; it consists in this: That in a sphere of very large di- 
mensions, and at distances from the surface very small in proportion to 
its radius, the part of the temperature independent of the time does not 
vary sensibly with these distances; and, that upon the normal at each 
point, whether at the surface or at an inconsiderable depth, it may be 
regarded as equal to the invariable part of the exterior temperature 
which corresponds to the same point. Hence it results, that the in- 
crease of heat in the direction of the depth which is observed near the 
surface of the earth cannot be attributed to the inequality of tempera- 
tures of different climates, and that it is necessary to look for the cause 
in circumstances which vary very slowly with the time. Whatever this 
cause may be, the difference of the mean temperatures of the surface 
and beyond, corresponding to the same point of the superficies, is pro- 
portional (according to a remark made by Fourier) to the increase of 
temperature upon the normal referred to the unity of length, so that 
