22 GANESHPRASAD, 



(a) If lim M{x, x') exists, it is equal to f(x). 



(b) If M(x,x') = M^{x, x')+ M^{x, x') such that lim J£ exists and 

 M^r^ COS \ where ilj{x')>^li--i], then the condition is that lim M —f(x). 



(c) M{x,x')r^cos [■^{x')\ where ip {x')';^l{^; further, f{x) = 0. 



iii. V(x, t) ceases to be the Solution, if at any point lim M{x, x') exists 



and is different from f{x). 



iv. V{x, t) ceases to be the Solution if , at every point within the interval 



{—71, 7t), ±D(x,x') either >•- 1 or x' '^"'"'"cos \il^(x')\ where ip{x')'^l 



Stable, unstable, and inadmissible initial states. 



23. I will call an initial state, T[x, 0) = f{^), stähle or unstable according 

 as both the conditions of Art. 10 are satisfied or only the first. An initial state 

 f{x) is considered inadmissible if it is not known that the first condition is satis- 

 fied. A stable initial state is called non-oscillatory if T{x, t) is continuous, i. e., 



lim V(x, f) — f(x) ; it is called oscillatory if there exists at least one point in 



the interval {—7t, 7t) where V{x, t) makes, within any indefinitely small interval 

 (0, t^), an infinite number of finite oscillations about f{x), i. e., T{x, t) has a dis- 

 continuity of the second kind at ^ = 0 and fix) is contained in the aggregate 

 of values assumed by V{x, t) as t approaches zero. 



A continuous initial state, if admissible, is always stable and non-oscillatory. 

 A discontinuous initial state, if admissible, may be stable or unstable, non-oscil- 

 latory or oscillatory : for example , if it satisfies ii. of the last article it is 

 stable; but it is non-oscillatory or oscillatory according as it corresponds to 

 one of the two (a), (b), or, for at least one value of x, to (c). If an admissible 

 initial state corresponds to iii. it is unstable. Remembering the conditions in- 

 volved in the integrability of f{x), it is easily seen that an unstable initial state 

 can be replaced by a stable one without changing V{x, t). 



An initial state is inadmissible if it corresponds to iv. For , Q {x, t) is in- 

 determinate and , consequently , it is not known whether the principle of the 

 conservation of energy , as stated in Art. 3 , is satisfied or not. It should be 

 noted that this is a case of failure not of mathematical analysis but of our 

 physical conceptions. 



Illustrative Examples. 



24. The following simple examples suffice to illustrate the salient features 

 of the theory : 



