CONSTITUTION OF MATTER AND ANALTTICAL THEOEIES OF HEAT. 21 



,•0 



V(x,t) behaves as f M{x,x')e äx' (II) 



where 



= f{X — x'), X = 7C. 



21. It follows from (II) that lim V{x, t) = lim M{x, x'). 



i = +0 x' == +0 



When lim M{x, x') does not exist , i. e. , when M{x, x') has a discontinuity 



of tlie second kind at a?' = 0, it would be sufficient for the purposes of this 

 essay to consider the case, M(x^ x')r^Q,o% \-^{x')\^ where ^{x')>^\. 



Casel: tl){x')-^l(^ 



X' (x') 1 1 

 Noting that ^ , ~ -y, where X(x') = , it is easily found by the 



methods of Arts. 14 — 17 that lim V(x, t) does not exist. 



i = +0 



Gase II: ip{x')>-ll~y 



It is easily found, by the method of Art. 18, that 



1 



^ ^ ^ 11 



\r{x, t)\^l, or ^[-r'^XiS/T)], i.e. according as ^l>{x')^-j, or l<-,- 



ip (yf ) XX 



Since — ^'(«')^-7) follows that \V{x, t)\ is always -^1. 



X 



Necessary and sufficient conditions for f(x). 



22. The group of conditions, of Art. 10, which is necessary and sufficient 

 in Order that V{x, t) be the Solution of the problem leads, with the help of the 

 results given in Arts. 19 and 21 , to certain necessary and sufficient conditions 

 for f{x), of an applicability sufficiently extensive for the purposes of this essay. 

 I give the simplest and most important of these conditions below. 



i. If f(x) is continuous in the interval {—tc, n) then , in order that V(x, t) 

 be the Solution of the problem , it is sufficient that there exist values of 

 x, forming an aggregate every where dense within the interval {—n, it), for 



which either | B {x, x')\<r^l or ± B {x, x') rsj x' ^'^'^ cos j (x') } where {x'} >~ 



Zc + v > 1. 



ii. Ii f{x) is discontinuous in the interval (—ff, n) then the condition rela- 

 ting to B {x, x'), given in i. , together with one of the following conditions is 

 sufficient : 



