CONSTITUTION OF MATTER AND ANALTTICAL THEORIES OF HEAT. 



3 



certain necessary and sufficient conditions for fix), which are of extensive appli- 

 cability. These conditions at once indicate the Classification of initial states as 

 stable or unstable, non-oscillatory or oscillatory, admissible or inadmissible. This 

 Classification is discussed in Art. 23. We conclude this part with Art. 24 in 

 which five examples ^) of initial states are given ; the Solution of the problem 

 corresponding to each of these being of Fourier' s type. In the first the initial 

 State is continuous, stable and non-oscillatory ; in the second, discontinuous but 

 stable and non-oscillatory; in the third, discontinuous, stable and oscillatory; 

 in the fourth as well as in the fifth, discontinuous and unstable. 



(4) 



In Part II. we give a carefuUy worked out theory which treats the solid 

 as molecular in structure but takes no account of the Constitution of the mole- 

 cule. The scheine of our exposition is as follows : — 



In Art. 25 we begin by specifying clearly and in detail the molecular 

 oscillations in the solid. The solid is thus supposed to contain rows of mole- 

 cules parallel to the axis of x; the molecular oscillations in each row being the 

 same. Each row contains r assemblages, the number of molecules in each assem- 

 blage being s. Now, with each assemblage in a row is associated a quantity 

 which is a function of time and which depends only on the molecular oscillations 

 in the assemblage; this quantity we call the temperature of the assemblage. At 

 the end of Art. 25 is given the following formulation of the problem: — 



The initial temperatures, T^{0), T^{0), ... T.(0), of the assemblages A^, Ä.^, 

 ... in any particular row being given , find their subsequent temperatures. 



The investigation embodied in Arts. 26—81 requires very delicate conside- 

 rations and is the most difficult portion of the exposition. In these Articles we 

 show how, by means of the hypotheses («J, (^J and (yj of Art. 27, approximate 

 analytical representations of the actual conditions of the phenomenon can be ob- 

 tained in terms of a continuous function Y{x, t) which we call the aiixiliary 

 function of the problem; this function is such that Y(x^^^, t) — T^{t) where x^ ,^ 

 is the - coordinate of the centre, and T^{i) the temperature, of the assemblage 

 at time t. As the first step to this end, we find in Art. 28 an approximation 

 to the quantity of heat which flows across a unit area, placed at right angles 

 to the axis of x, in any interval {t, t-\-t): the result is given in the equation (I). 

 As the next step we obtain in Art. 29 an approximation to the quantity of heat 

 absorbed by a cylinder, with its axis parallel to the axis of x and its faces, of 

 unit area, x = x^, x — x^, in any interval : the result is given in the equation (II). 



1) It is liardly necessary to mention that the functions f(x) used in the examples (iii) and 

 (iv) belong to a new type which was suggested to the present writer by the procedui-e in Arts. 

 110* — 11* of Dini-Lüroth's „Gruüdlagen für eine Theorie der Functionen einer veränderlichen 

 reellen Grösse." The investigation given there is, with evident modifications , applicable to these 

 functions. 



1* 



