M. POISSON ON THE MATHEMATICAL THEORY OF HEAT. 133 



general case by a quadruple integral, whicli 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 vEilue 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 formulae. 



Chapter XI. O71 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 obsei-ved 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 verj' 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 lengtli, so that 



