of the Mechanical Theory of Heat. 169 



In the determination of the second quantity, we must, on 

 account of the gradual nature of the increase pY, conceive the 

 factor jii divided into an infinite number of parts, and for each 

 part reckon as the initial value of V that which corresponds to 

 the place where the moveable point was at the moment when 

 this part commenced. Considering thus the part fjbd<f), which 

 has arisen during the phase-element from <p to <f> + dcj), we have 

 to form for it as expression of the work the difference 



^(V,-V), 



in which V and V 1 denote those function-values which belong to 

 the phases cf> and (f) { . Properly the variations of the function- 

 values would also have to be taken into account, because the 

 moveable point is, from the beginning of alteration of the force, 

 no longer in the original path. As, however, these variations 

 are infinitesimal and the factor fju is also infinitely small, only 

 infinitely small quantities of a higher order would hence arise, 

 which may be neglected. In order, then, to extend to the whole 

 increase the above expression, which is valid for an infinitesimal 

 part of it fjN, we must integrate it from to 1. By resolving 

 the parenthesis the expression is divided into two terms. The 

 first gives /ubV^dcj) by integration, or, since Vj is independent of 

 (j>, simply jxVj. The integral of the other term, fjtNdcj), can be 

 represented by fiV, if V denote the mean value of V during an 

 entire revolution. Accordingly the second quantity sought is 



By adding the two quantities, we obtain the variation of work 

 corresponding to the phase </> p namely 



8U l+ MV,-V). 

 In order to deduce, further, the work SL, which refers to the 

 whole alteration of the stationary motion, we must multiply this 

 expression by d<fi l} and once more integrate it from to 1. We 

 thus obtain 



8L=\ SU^ + fi) (Y l - V) dcf> l} 



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for which, since in the first term on the right-hand side the in- 

 tegral of the variation may be replaced by the variation of the 

 integral, we may write 



8L=&f 1 U 1 ^ 1 + ^f 1 (V 1 -V)# 1 . 



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The integrals I V l d(j) ] and /ju 1 V l dcj> l signify the mean values of 

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