M. POISSON ON THE MATHEMATICAL THEORY OF HEAT. 13i 
xem, by means of which we reduce a numerous class of double integrals 
to simple integrals. 
Cuarter IX. Distribution of Heat in.a Bar, the transverse Dimensions 
of which are very small—We form directly the equation of the motion 
of heat in a bar, either straight or curved, homogeneous or heterogeneous, 
the transverse sections of which are variable or invariable, and which 
radiates across its lateral surface. We then verify the coincidence of 
this equation with that which is deduced from the general equation of 
Chapter IV., when the lateral radiation is abstracted and the bar is cy- 
lindrical or prismatic. This equation is first applied to the invariable 
state of a bar the two extremities of which are kept at constant and 
given temperatures. It is then supposed, successively, that the extent 
of the interior radiation is not insensible, that the exterior radiation 
ceases to be proportional to the differences of temperature, that the ex- 
terior conductibility varies according to the degree of heat, and the 
influence of those different causes on the law of the permanent tempera- 
tures of the bar is determined. Formule are also given, which will 
serve to deduce from this law, by experiment, the respective conducti- 
bility of different substances, and the quantity relative to the passage from 
one substance into another, in the ease of a bar formed of two heteroge- 
neous parts placed contiguous to and following one another. After 
having thus considered in detail the case of permanent temperatures, we 
resolve the equation of partial differences relative to the case of va- 
riable temperatures; which leads to an expression of the unknown quan- 
tity of the problem, in a series of exponentials, the coefficients of which 
are determined by the general process indicated in Chapter VII., what- 
eyer may be the variations of substance and of the transverse sections 
of the bar. We then apply that solution to the principal particular 
cases. When the bar is indefinitely lengthened, or supposed to be 
heated only in one part of its length, the laws of the propagation of heat 
on each side of the heated place are determined; this propagation is in- 
stantaneous to any distance; a result of the theory presenting a real 
difficulty, but the explanation of which is given. 
CuarrTer X. On the Distribution of Heat in Spherical Bodies —The 
problem of the distribution of heat in a sphere, all the points of which 
equally distant from the centre have equal temperatures, is easily brought 
to a particular case of the same question with regard to a cylindrical 
bar. It is also solved directly; the solution is then applied to the two 
extreme cases, one of a very small radius, and another of a very great 
one. In the case of an infinite radius, the laws are inferred of the pro- 
pagation of caloric in a homogeneous body, round the part of its mass 
to which the heat has been communicated, similarly in all directions. 
We then determine the distribution of heat in a homogeneous sphere 
K 2 
= 
