CONSTITUTION OP MATTER AND ANALTTICAL THEORIES OF HEAT. 67 



An important feature of the theory is the distinction between admissiUe 

 and inadmissible initial states. For example, let ^{x) = 6,(x) where T{x, 0) = ^.{x) 

 is an inadmissible initial state; further, let V^{x,t) be the function V{x,t) corre- 

 sponding to 9. {x). Also let V[ be the greatest value that | F. {x, t) \ can have. 

 Then the description embodied in the equation 



T{x,t) = V,{x,t) + ^V,(x,t), ^>0, 



i 



is inconsistent and, consequently, untrne ; although the result of the Observation 

 of the initial temperature is still truly described by the equation 



T(x,0) = x'' + e%,{x). 



The Discontinuous Theory. 



64. The importance of the theory, worked out in Part II., lies in the fact 

 that, in all the cases that can possibly come under our Observation, it con- 

 sistently and truly describes the phenomenon by aid of an old and familiär 

 mechanism: it may, therefore, be regarded as the simplest of the type of 

 theories which, in their analysis of the Constitution of the solid, stop short 

 at certain finite entities , called atoms or molecules , simply regarding them as 

 structures whose Constitution is inscrutable. The essential feature of the theory 

 is that its description of the phenomenon is professedly inexact. 



The Improperly Continuous Theory. 



65. The theory, worked out in Part III., consistently and truly describes 

 the phenomenon in terms of the characteristic function. And in width of scope 

 it is scarcely inferior to the continuous theory. Thus , in any particular case, 

 it can describe the phenomenon in an infinite number of tvaijs , the initial cha- 

 racteristic function being assumed to be continuous or discontinuous , stable or 

 unstable, oscillatory or non-oscillatory. 



The theory is marked out from the continuous and the discontinuous 

 theories by two important features. One is that, although it does not regard 

 the solid as a continuum with the same properties in all its points, its de- 

 scription of the phenomenon is not only consistent and true but also thorough- 

 going and exact. The other feature is that the hypotheses on which it is based 

 admit of an Interpretation which is not in disagreement with our physical con- 

 ceptions. This, of course, cannot be affirmed of the continuous theory; and 

 even in the discontinuous theory the hypothesis that conduction is whoUy due 

 to molecular radiations is questioned by high authorities ^). 



1) See Larmor's Memoir (Phil. Trans. Vol. 190), Art. 11. 



