CONSTITUTION OF MATTER ÄND AJSTALTTICAL THEORIES OF HEAT. 



5 



the fifth example this change is so great that the theory fails to give any 

 approximation to the Solution. 



In Arts. 40 — 43 we give a brief criticism of Fourier' s theory. We begin 

 by pointing out that Fourier' s theory is a continuous one ; we then find in Art. 

 41 the conditions which are necessary and sufficient in order that the conditions 

 given by Fourier's theory may have any meaning and, further, foUow from the 

 conditions (A), (JB) and (C) of our theory. In Art. 42 we consider the Solutions 

 of Fourier's type and prove the following result : — 



In order that the conditions given by Fourier's theory viz., 



^ 0 = ^ t), 



T{x,t) is continuous in t, and T'{n,t)=^Q, T'(~7t,t) = 0, —7r<.x<7t, 0<t, 

 may have meaning and necessarily follow from (A), (B) and (C), it is necessary 

 that /■" (x) exist and be finite ; and it is sufficient that f" (x) be finite and con- 

 tinuous, and / '(jt), f {—Tt) be zero. 



We conclude this part with Art. 43 which contains seven examples specially 

 selected to illustrate the limited scope of Fourier's theory. 



(5) 



Part III. contains a theory which regards the solid as improperly continuous ; 

 to work out this theory we have to employ an improperly continuous analysis, 

 i. e. , an analysis in which one of the independent variables has for its domain 

 an everywhere dense but enumerable aggregate. "We begin by carefully speci- 

 fying in Art. 44 the notation which we employ. In Art. 45 we formulate the 

 foUowing improperl}^ continuous theory of solids : 



With the slab is associated an enumerable aggregate G, of positive numbers, 

 which is everywhere dense in the interval (0, tc) and contains ä: we call Gr the 

 discriminating aggregate of the slab. Also, two slabs diflFer only in this that they 

 have different discriminating aggregates. Taking § for a variable with the aggre- 

 gates — G and G as its domain, we define the temperature T{x, t) at any point 

 X by the equations 



^ V X — 1 



where l is dependent on G and is less than, say, 10"'. 



These are the equations of page 50: i^) is called the charaderistic 



function of the slab. At the end of Art. 45 the problem of linear conduction is 

 thus formulated : 



