58 



GANESH PRASAD, 



f{x) is continuous at |. Therefore %{i„ t) is the characteristic fanction of the 

 problem and the initial state, C(|, 0) = is stable and non-oscillatory. 



(ii) . Suppose tbat the discriminating aggregate G of the slab is {it, R^). 

 Let 



and CO,,, having the same meanings as in (iii) of Art. 24 and G being taken to 

 be identical with the sub-aggregate represented there by G^ ; then qo (|) possesses 

 an associate f{x) = f\{x), f^{x) standing for the fix) in (iii) of Art. 24. 



At any point |, | F' (x, _ ^ | nnj 1 ; also xi^^t) makes , within any inde- 



finitely small interval (0, f j , an infinite number of oscillations , about cp (|) , of 



2 



finite amplitude not greater than -^^^ • 



Therefore % % t) is the characteristic function of the problem and the initial 

 State, C(|, 0) = qo(|), is stable and oscillatory. 



(iii) Suppose that the discriminating aggregate G of the slab is (je, 

 Let 



' ?P"j and (On^ having the same meanings as in (iv) of Art. 24 and G being taken 

 to be identical with the sub-aggregate G^ = | oj»,. | ; then cp (^) possesses an asso- 

 ciate f(x) — f\{x), f^{x) standing for the f{x) in (iv) of Art. 24. 



%{l,t) is the same as in (ii). Since *P"j(0)>2, (p{l) is not contained in the 

 aggregate of values assumed by %(|, t) as t approaches zero. Therefore the 

 initial state is unstable and, so to speak, runs down instantaneously to the ini- 

 tial State of (ii). 



(iv) Suppose that the discriminating aggregate G of the slab is (tv, i?,), 



M 



standing for the aggregate of all the rational numbers with an integral power 

 of 3 as denominator, in the interval (0, jt). 

 The initial states 



1 " 



C{^, 0) = "'T S^l, I > 0, 



1 



1 



are, respectively, stable and non-oscillatory, stable and oscillatory, and unstable. 



