M. POISSON ON THE MATHEMATICAL THEORY OF HEAT. ISl 



rem, by means of which we reduce a numerous class of double integrals 

 to simple integrals. 



Chapter 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 jjermanent tempera- 

 tures of the bar is determined. Formulae 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 case 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 cxpressionof 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- 

 ever may be the variations of substance and of the transverse sections 

 of the bar. We then apply that solution to the principal particular 

 cases. WTien 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 theorj^ pi'esenting a real 

 difficulty, but the explanation of which is given. 



Chapter 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 conniiunicatod, similarly in all directions. 



We then determine the distribution of heat in a homogeneous sphere 



K 2 



