42 



GANESH PRASAD 



2 



(v) Suppose that = 10"' A^, — = X^, and A = ö"' x 10"' x X[. Let r be 

 odd ; and, furtlier, let the a;-coordinates , „, a^^.o» • • • ■^r.o ; of the initial positions 

 of the centres of the assemblages be given by the zeroes of j + tan 

 where Stands for the sub-multiple of je nearest to, say, •75xlO""XA,. Thus 



Xr+\ ^ = 0, 



2 



— |— + ?. 0 Tt 



a;'*' standing for the ^th positive root oi x+ tan x = 0. Now, let the initial 

 auxiliary function Y{x,Q) = 1 + it; sin j. Then, the left side of (PJ is greater 



tban , > 8 X 101 Therefore, however small A be, the theory falls to 



give any approximation to the Solution of the problem, ever so crude. 



Criticism of Fourier' s Theory. 



40. Comparing Fourier' s theory with the theories expounded in the pre- 

 ceding pages, it is readily seen to be a continuous one. I will conclude this 

 part with a hrief criticism of Fourier's theory pointing out its limited scope. 



Starting with the same conditions of the phenomenon as in the preceding 

 part and with — essentially — the same hypotheses as (a) and (/3), Fourier's 

 theory ^) gives the following analytical expressions of the conditions of the phe- 

 nomenon : 



Ar(.,o = ^rfeO. (Ä) 



T {x, t) is continuous in t, and (B) 



T'{'jt,t) = 0, r(-7E,o = 0. (C) 



It should be noted that these conditions are required by Fourier's theory 

 to hold for 0<lz;, —n<x<%] also, in order that they have any meauing, it 



d 



is necessary and sufficient that ~T{x,t) and -^T{x,t) exist and be finite. 

 Therefore (B) is involved in the statement of (A). 



1) „Theorie Analytique de la Clialeur", Chapter II. See specially Arts. 117 aud 120. 



