36 Prof. R. Clausius on the Theorem of the Mean Ergal, 

 we get 



l i= A( c " a ^[ 1 ~a(^T)- 2.4(k-i) --0}- {U) 



Multiplying equations (21) and (22) together, we obtain 

 l/dx\ 2 ?, dx / ] a» 



2\dt) dt= v^V k -{c^xr' 



If this equation be treated in the same way as (22), taking into 

 consideration that we can put 



1 f *7« fe? Y^ 1 (dxX* 1 . 

 U \dt) dt =2\di)'V> 



and, for abbreviation, we introduce w with the signification 



-if/' ^ 



we arrive at the following one : — 



1 . /if »r, 1 1.1 



l m TVA C ~ ak "L 1 + 2(^i) + 3.4(3»-l) 



^S^bl)*'"]} <*» 



As, finally, if we denote by mh the mean value of that part of 

 the ergal which relates to the x- direction, we may put h^=k — w, 

 we obtain from (24) and (26) : — 



\M= A 8 ^[^4. _L_ + a ^_ x) + . . .]. (a7 j 



To simplify equations (24), (26), and (27), we will substitute 

 a single letter for the series which occurs in (26), putting 



R=1+ 3^1) + 2 .4(an-l) + a.4.6(3«-l) + ' •• (38) 



? , o 



When this equation is multiplied by and then the follow- 

 ing quantity developed in the form of a series, 



2 2/ 1 1.1 1.1.3 \ 



rc^ 1 - 1 ^ 2~^~2~4~6~'"--7' 

 which is =0, is added, we get 



ti-2 r __ 1 1.3 1.3.5 



n 2(n-l) 2.4(2^-1) 2.4.6(3/1—1) ' 



the series which occurs in (24). From this it results simulta- 



