CONSTITUTION OF MATTER AND ANALTTICAL THEOEIES OF HEAT. 29 



the cyliiidrical volume, occupied by such assemblages, is, at any time in the 

 interval, less than, say, 



Oll A , V 



wliere represents the greatest length that an assemblage can have. 

 Let represent the part of H absorbed by these assemblages. 

 Then 



I I < c |2iA, + ^ (a;, - a; j| ß ^ + r), 



where Sl{t,t + T) stands for the greatest value of \Y(x, t+t) — Y(x,t)\. Bat 

 evidently Spl, > 2^^^ + X\ Therefore 



= 3pk^%c£l{t,t+T), I e, I < 1. 



Consider = H—H^ the quantity of heat absorbed by those assemblages 

 which remain wholly inside the cylinder throughout the interval. 



Let ^,„1+1 , Ahi+2 , • • . -4mi+n , • • • •4»»,+» be a row of these assemblages ; 

 fnrther, let 



be the faces of A„,^+r^ at time ^ + -2' -^n being greater than X,.j, Then it 

 foUows from (yj that 



i'l =1 



where 6^ is the sum of the areas of the bases of the rows. 

 Consider 



I = c/""' I Y(x', t + t)- Y(x', t) I äx'. 



Now, 



Ijc = 2lX,%Sl{t,t + t)+ 2 [/ \Y(x',t + T)-Y(x',t)\dx'] 



'■1 = 1 X 



[f '''^'\Y{x',t + v)-Y(x',t)\dx'], 



Let 'il(x„x^) stand for the greatest of the values of the fluctuation of 

 Y{x',t) in intervals, of length l^, taken any where within the interval (x^,x^); 



further, let tl, represent 1--^ where evidently depends on the position 



of L in the plane containing it, and the size and shape of the periphery of R 

 Then, noting that 



