Intr o duction. 



(1) 



The object of this essay is to show how the recent advances in mathema- 

 tical analysis may be used for working out analytical theories of the linear 

 conduction of heat in a homogeneous solid; each theory being based on definite 

 suppositions as to the Constitution of the solid. The essay is divided into four 

 parts. In Part I. we give a theory which treats the solid as a contiuuum with 

 the same properties in all its points. Part II. contains a carefully worked out 

 theory which treats the solid as molecular in structure but takes no account of 

 the Constitution of the molecules; at the end of this part a criticism of Fou- 

 rier' s theory is given. In Part III. is worked out a theory which regards the 

 solid as improperly conUnuous, i. e., as a continuum in which an everywhere dense 

 but enumerable aggregate of points is marked out to distinguish the solid from 

 all other solids ; this part concludes with a discussion of the question of the 

 „uniqueness of the Solution." Part IV. contains a brief discussion of the theories 

 expounded in the previous parts. 



(2) 



We restrict ourselves throughout this essay to the simple problem of linear 

 conduction in an infinite slab bounded by two parallel planes impermeable to 

 heat and at distance 2n from each other, the initial temperature being an even 

 function of the distance from the central plane of the slab. The first step to- 

 wards the Solution of this problem is to define the conditions of the phenomenon. 

 We take the fundamental condition to be the satisfaction of the principle of the 

 conservation of energy; the other condition is that if with varying tivie any 

 physical quantity passes from one value to another it assumes all the inter- 

 mediate values. The next step is to transform the problem from a physical into 

 an analytical one; and this is done by translating, with the help of suitable 

 hypotheses, the above two conditions, and the supposition of the impermeability 

 of the faces, into analytical language. We then find without difficulty an ex- 



Abhaualg. d. K. Ges. d. Wiss. zu Göttingen. Math.-Pliys. Kl. N. F. Band 2, 4. 1 



