132 M. POISSON ON THE MATHEMATICAL THEORY OF HEAT. 



covered with a stratum, also homogeneous, formed of a substance differ- 

 ent from that of the nucleus. During the whole time of cooling, the tem- 

 perature of this stratum, however small its thickness may be, is differ- 

 ent from that of the sphere in the centre, and the ratio of the tempera- 

 tures of these two parts, at the same instant, depends on the quantity 

 relative to the passage from one substance into the other, of which we 

 have already spoken. From this circumstance an objection arises 

 against the method employed by all natural philosophers to determine, 

 by the comparison of the velocities of cooling, the ratio of the specific 

 heat of different bodies, after having brought their surfaces to the same 

 state by means of a very thin stratum of the same substance for all 

 these bodies. The quantity relative to the passage of the heat of each 

 body in the additive stratum, is contained in the ratio of the velocities 

 of cooling ; it is therefore necessary that it should be known in order 

 to be able to deduce from this ratio, that of their specific heats. A 

 recent experiment by M. Melloni proves that a liquid contained in a 

 thin envelope, the interior surface of which is successively placed in dif- 

 ferent states by polishing or scratching it, always cools with the same 

 velocity, whilst tlie ratios of the velocity change very considerablj% 

 as was known long before, when it is the exterior part of the vessel 

 that is more or less polished or scratched. The quantity relative to the 

 passage of caloric across the surface of separation of the vessel and the 

 liquid, is therefore independent of the state of that surface, a circum- 

 stance which assimilates the cooling power of liquids to that of the 

 stratum of air in contact with bodies, which in the same manner does 

 not depend on the state of their surface, according to the experiments 

 of MM. Dulong and Petit. 



When a homogeneous sphere, the cooling of which we are consider- 

 ing, is changed into a body terminated by an indefinite plane, and is 

 indefinitely prolonged on one side only of that plane, the analytical ex- 

 pression for the temperature of any point whatever changes its form, in 

 such a manner that that temperature, instead of tending to diminish in 

 geometrical progression, converges continually towards a very different 

 law, which depends on the initial state of the body ; but however great a 

 body may be, it has always finite and determined dimensions; and it is al- 

 ways the law of final decrease enunciated in ChapterVI. which it is neces- 

 sary to apply ; even in the case, for example, of the cooling of the earth. 

 If the distribution of heat in a sjjhere, or in a body of another form, 

 has been determin( d, by supposing this body to be placed in a medium 

 the temperature of which is zero, this first solution of the problem may 

 afterwards be extended to the case in which the exterior temperature va- 

 ries according to any law whatever. In my first Memoir on the theory of 

 heat, I have followed, in regard to this part of the question, a direct me- 

 thod applicable to all cases. According to this method, one part of the 

 value of the temperature in a function of the time is expressed in the 



