Page:Scientific Memoirs, Vol. 1 (1837).djvu/143

 Rh, by means of which we reduce a numerous class of double integrals to simple integrals.

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 cylindrical 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 exterior conductibility varies according to the degree of heat, and the influence of those different causes on the law of the permanent temperatures of the bar is determined. Formulæ are also given, which will serve to deduce from this law, by experiment, the respective conductibility of different substances, and the quantity relative to the passage from one substance into another, in the case of a bar formed of two heterogeneous 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 variable temperatures; which leads to an expression of the unknown quantity of the problem, in a series of exponentials, the coefficients of which are determined by the general process indicated in Chapter VII., whatever 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 instantaneous to any distance; a result of the theory presenting a real difficulty, but the explanation of which is given.

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 propagation 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 Rh