M. POISSON ON THE MATHEMATICAL THEORY OF HEAT, 125 
formed of the same material. After having considered the influence 
of the air upon radiation which we had at first eliminated, we give at 
the end of this chapter a formula which expresses the instantaneous 
variations of temperature of two material particles of insensible magni- 
tude, by means of which the exchange of heat takes place after one or 
many reflexions upon the surfaces of other bodies through air or through 
any gas whatever. 
Cuarter III. The Laws of Cooling in Bodies having the same Tem- 
perature throughout.—While a homogeneous body of small dimensions 
is heating or cooling, its variable temperature is the same at every 
point; but if the body is composed of many parts formed of different 
substances in juxtaposition, they may preserve unequal temperatures 
during all the time that these temperatures vary, as we show in an- 
other chapter. Inthe present we determine, in functions of the time, 
the velocity and the temperature which we suppose to be common to 
all the points in a body placed alone in a sphere either vacuous or 
full of air, and the temperature of which is variable. If the sphere 
contains many bodies subject to their mutual influence upon each other, 
the determination of their temperatures would depend on the integra- 
tion of a system of simultaneous equations, which are only linear in the 
case of ordinary temperatures, but in which we cannot separate the 
variables when we investigate high temperatures, and when the radia- 
tion is supposed not to be proportional to their differences. 
Experiment has shown that in a cooling body, covered by a thin 
layer or stratum of a substance different from that of which it is itself 
composed, the velocity of refrigeration only arrives at its maximum when 
the thickness of this additive stratum, though always very small, has 
notwithstanding attained a certain limit. We develop the consequences 
of this important fact in what regards extension of molecular radiation, 
and explain how those consequences agree with the expression of the 
passage of heat found in the preceding chapter. 
Cuarrter IV. Motion of Heat in the Interior of Solid or Liquid 
Bodies.—We arrive by two different processes at the general equation of 
the motion of heat; these two methods are exempt from the difficulties 
_ which the Committee of the Institute, which awarded the prize of 1812* 
to Fourier, had raised against the exactitude of the principle upon which 
his method was sustained. The equation under consideration is appli- 
- cable both to homogeneous and heterogeneous bodies, solid or fluid, at 
rest or in motion. It was unnecessary, as they appeared to have 
: | thought, to find for fluids an equation different from the one I ob- 
__ * This Committee consisted of MM. Lagrange, Laplace, Legendre, Haiiy and 
i Malus. 
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