90 REPORT 1891. 



Here the terms 



represent the woi-k done on the system by variation of the controllable 

 coordinates — i.e., the external work performed on the system. Hence, if 

 8W denote the external ivorlc performed by the system, as in § 2, we have 



^l^Bp=-8W (8) 



Substituting in equation (5) from (6), (7), (8), in succession, we have 



8\l'2T:dt=[^m(x8x + ySy + iSz)T+(\8T + SY + 8W)dt . (9) 



But if 8Q represents the variation of energy communicated through the 

 molecular or uncontrollahle coordinates, we have, by the Principle of Con- 

 servation of Energy' (equation 1), 



8Q=SE + SW = ST + 8Y + SW. 



Therefore (9) gives 



z['2Tdt=\^m{x8x-Vy8y + ~h)T+[^8qdt . . (10) 



Let t2 — ti=i, and let mean values with respect to the time be indi- 

 cated in the usual way by a vinculum drawn over them, then the last 

 equation (10) may be written 



8(2iT) = [^m(x8x + ySy + i8z)T '+i.Jq -. . (11) 



whence 



8Q 8(2iT)_ [^^"^"^^-'' + ^^^ + ~'^^U' ■ . . (12) 

 T~ iT ~ /T 



Hence, if we assume the quasi-period i to be defined, as postulated 

 (assumption 1), by the relation 



[^i«0tS.« + 7/S2/-H~S.)]''=O. . . . (13) 

 =S21og (iT) = Slog(iT)2 . . (14) 



we shall have 



8Q_ S(2iT ) 

 T~ tT 



13. Equation (14) is analogous to the thermodynamical equation (2) 

 when written in the form 



o 



the mean kinetic energy of the molecules T taking the place of the 

 absolute temperature 6. 



Thus Carnot's principle is proved for reversible transformations, pro- 



' This step was omitted by Szily, who fell into several errors in consequence, and 

 it is not explicitly mentioned in Clausius' writings. 



