covered with a stratum, also homogeneous, formed of a substance different from that of the nucleus. During the whole time of cooling, the temperature of this stratum, however small its thickness may be, is different from that of the sphere in the centre, and the ratio of the temperatures 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 different states by polishing or scratching it, always cools with the same velocity, whilst the ratios of the velocity change very considerably, 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 circumstance 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 considering, is changed into a body terminated by an indefinite plane, and is indefinitely prolonged on one side only of that plane, the analytical expression 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 always the law of final decrease enunciated in Chapter VI. which it is necessary to apply; even in the case, for example, of the cooling of the earth.
If the distribution of heat in a sphere, or in a body of another form, has been determined, 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 varies 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 method 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
