CONSTITUTION OF MATTEE AND ANALTTICAL THEOEIES OF HEAT. 



7 



P»art I. 

 Continnous Theory'). 



Formulation of the Problem. 



1. In the present part I propose to examine the phenomenon of the linear 

 conduction of heat in a homogeneous solid from the point of view of the con- 

 tinuous theory of matter ; I will, therefore, treat the slab as a conünuum with 

 the same properties in all its points. 



Analytical representation of the conditions of the Phenomenon. 



2. Take a straight line perpendicular to the faces of the solid as the 

 axis of X. Let the faces be given by 



X = —Tt , X = +7r. 



Let t be another continuous variable ; and, unless the contrary is mentioned, 

 suppose that 0<^, — ti ^ x < 7t. 



Then the temperature of any point of the solid at time ^ is a funetion of 

 X and t only. Let T{x, t) be this funetion. 



3. Let S be any closed surface in the solid. Then the principle of the 

 conservation of energy requires the equality of the quantity of heat which 

 flows into S in any interval t + x) and the quantity of heat absorbed by the 

 enclosed solid in the same interval. It is evident that the principle is satisfied 

 if the equality is proved for the case when S is any cylindrical surface with 

 the axis of x as its axis. 



1) In working out this tlieory I have received very great lielp from Fourier's „Theorie 

 Analytique de la Chaleur". I am also greatly indebted toDu Bois Reymond's „Untersuchungen 

 über die Convergenz und Divergenz der Fourierschen Darstellungsformeln" (Abhandlungen der k. 

 bayer. Akademie, Bd. XII). 



