Therefore 



Now 



■where 



Therefore 



CONSTITUTION OF MATTER AND ANALTTICAL THEORIES OP HEAT. 13 



V'ix,t) = f \f(x + x')-f{x-x')] &' {x')äx' + r(t). 



(2?r— g)' 



X' 



"where 



Let D(x, x') stand for f(^ + ^')-Ji^- ^ . Then, 



ff — — 



V'ix.t) behaves as ;= f D(x,x')x'^e dx' (T) 



wlien t approaches zero. 



13. If \D {x, x')\ < Ä , a finite quantity, as x' approaches zero, then it 

 foUows from (I) that 



I V (x, t) I < — ^ f x"e dx' 



as t approaches zero. x'^e dx' approaches 1 as ^ approaches 



2 \jtt X t'o 



zero. Therefore \ V' {x, t)\ < Ä as t approaches zero. 



If D {x, x') becomes indefinitely large as x' approaches zero, then it is easily 

 Seen that V {x, t) becomes indefinitely large, as t approaches zero , and has the 

 same sign as Z) [x, x'). In particular, when D (x, x') oj x"'"'^ as x' approaches zero, 



o<^<i, v'{x, t)o^{\/Vy-\ 



I proceed now to consider the case when D (x, x') makes an indefinitely 

 large number of fluctuations, with indefinitely increasing ampUtude, as x' appro- 

 aches zero. It would be sufficient for the purposes of this essay to investigate 



