CONSTITUTION OP MATTER AND ANALYTICAL THEORIES OF HEAT. 



55 



Stable, unstable, and inadmissible initial states. 



53. I will call an initial state, C 0) = 9 (^), stable or unstable according 



as both the conditions of Art. 51 are satisfied or only the first. An initial 



State 9 (I) is considered inadmissible if it is not known that the first condition 



is satisfied. A stable initial state is called non-oscillatory if C (|, t) is continuous, 



i. e., lim % (|, t) = q) (5) ; it is called oscillatory if there exists at least one 

 t = +o 



value of I for which % t) makes, within any indefinitely small interval (0, /"J, 

 an infinite number of finite oscillations about (p (^), i. e. , C(|. t) has a discon- 

 tinnity of tbe second kind at ^ = 0 and cp (|) is contained in the aggregate 

 of values assumed by % 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- 

 oscillatory 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 |, to (c). If an admissible 

 initial state corresponds to iii. it is unstable. ßemembering the conditions involved 

 in the supposition that q}(^) possesses an associate, it is easily seen that an 

 unstable initial state can be replaced by a stable one without changing x (I, 0- 



An initial state is inadmissible if it corresponds to iv. For Q (§, t) is 

 indeterminate, 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 the 

 hypotheses we started with. 



Approximations to impossible initial states. 



54. Let Ti (j) stand for an even function of x such that there does not 

 exist any even function which can be connected with T^^x) by the equations 



(^) = irf^^G (^) d^,0^x<7c-l, 



x—l 



1 r"^ 



7C-X+1 



Then (x) cannot represent a temperature ; and, therefore, an initial state 

 in which the temperature is supposed to be T^{x) must be an impossible one. 

 I proceed now to prove the foUowing theorem: 



If T- (./:) is a continuous function of x and d an arbitrarily small but fixed 

 qiiantity^ then it is ahvays possible to find an admissible cliar acter istic function C (|, 0) 

 such that, for all the poiyits in the interval (l, ä — l), the difference between T^{x) 

 and T {x, 0) is mimerically less tltan d. 



