66 



GANESH PEASAD, 



and is therefore essentially inexact; finally, the third is thorough-going and 

 exact althongh it does not regard the solid as a continuum with the same pro- 

 perties in all its points. 



The Continuous Theory. 



63. The theory, worked out in Part I., is distinguished from Fourier's 

 theory not only by its ivider scope hut also by its greater siniplicity ] for its 

 description of the actual conditions of the phenomenon is recognised as a true 

 one from the very form of the description : it is therefore the simplest possible 

 continuous theory. 



The essential feature of the theory is that, in any particular case, it can 

 describe the phenomenon in an infinite nuiiiber of ivays , the initial temperature 

 being assumed to be continuous or discontinuous, stable or unstable, non-oscilla- 

 tory or oscillatory: and all these descriptions will be consistent and true. The 

 precise import of this remark is made clear by the following example : 



Suppose that the result of the Observation of the initial temperature of 

 the slab is embodied in the statement that 



T {x, 0) ^ f{x) = + e 9 {x), (1) 



where e represents the greatest possible error of Observation and 0 < 1 6 (a;) | < 1. 

 Now, let T^{x,t), T^{x,f), T^{x,t), T^{x,t) and T^{x,t) stand for the T{x,t) in 

 the first, second, third, fourth and fifth examples of Art. 24, respectively ; also 

 let T' be, numerically, the greatest value that any of the five T's can have. 

 Then , representing by T^{x,t) the T{x,t) corresponding to T(x,0) = x'^, the 

 descriptions embodied in the equations, 



and 



Tix, t) 











T{x, t) 







e 



~r 



T,ix,t), 



T{x, t) 





t) + 



e 



TAx,t), 



T(x, 0 



= [x. 



,t)+ 



e 



~r 



TAx,t\ 



T{x, t) 



= T,{x, 



, 0 + 



e 

 -jr 



T,{x, t), 



T{x, t) 



= (x. 



,0 + 



e 



~T 



T,(x, 0, 



are all consistent and true. And, in each of the first two of these descriptions, 

 the initial temperature is continuous , stable and non-oscillatory ; in the third, 

 discontinuous, but stable and non-oscillatory ; in the fourth, discontinuous, stable 

 and oscillatory; and in the fifth as well as in the sixth, discontinuous and 

 unstable. 



