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

 My equation then reads, 



g(U-f)=2^'S.logi+S— Be, . . . (I.) 



in which the first summation on the right-hand side refers to 

 all the n variables q, and, consequently, to all the n intervals i, 

 and the second summation applies to the above-mentioned con- 

 stants occurring in the ergal. 



In my previous memoir I have further shown that this equa- 

 tion not merely holds good when the quantities q l9 q 2 , . . . q n occur 

 periodical^, but is also applicable to other stationary motions, if 

 the time-intervals i v i 2 , . . . i n can be chosen so as to satisfy a 

 certain condition-equation there given, into the consideration of 

 which, however, I shall not here enter, as it would require ex- 

 planations which are not needed for the understanding of what 

 follows. 



§2. To the preceding equation other forms can be given 

 which are theoretically interesting and also convenient to use. 



From equation (2) we get 



2ST=2% 7 (3) 



If this equation be added to (I.), and E be put for the sum 

 U+ T, which denotes the energy of the system, we then obtain 



SE = 2ls{f q 'i)+X d -^Sc, .... (II.) 

 or, in another form, 



5E=!£^S!og(^) + £^Sc. . . (II A .) 



Dividing equation (3) by 2, and then adding it to (I.) ? gives 



8U-JsJ*(^+S^&, . . (III.) 

 or, in another form, 



BV=^2fq'B\o S {^)^^Bc t . (IIIa.) 



In equation (I.) the quantity U — T is to be considered as a 

 function of the various intervals i and constants c; and the 

 equation can be analyzed into as many partial equations as there 

 occur independent variations on the right-hand side. Just so 

 in equations (II.) and (IIa.) the energy is to be looked upon as 

 a function of the several quantities pq'i and the constants c, and, 

 finally, in equations (III.) and (IIIa.) the mean ergal U as a 

 function of the quantities pq'i 2 and the constants c. 



