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 definite 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 formulæ.
Chapter 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 chapters 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 employed 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 solution of the problem the most important is that for which we are indebted to Laplace; it consists in this: That in a sphere of very large dimensions, 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 increase of heat in the direction of the depth which is observed near the surface of the earth cannot be attributed to the inequality of temperatures 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 proportional (according to a remark made by Fourier) to the increase of temperature upon the normal referred to the unity of length, so that
