170 Prof. R. Clausius on the Second Axiom 



Uj and Vj during one revolution, or, which amounts to the 

 same, the mean values of U and V during one revolution, which 



are denoted by U and V. The integral i Ve^ is likewise 



Jo 



equal to V ; and consequently 



SL=SU+yu,(V-V)=oTL 



We have thus for this case also arrived at the same simple 

 result which we have already expressed for the other cases in 

 equation (20). 



To obtain this result, we have made the special assumption 

 that the alteration of the ergal proceeds uniformly during one 

 entire revolution. But we may also extend this result to an- 

 other case, and one which is important for the following. We 

 will imagine that, instead of one point in motion, there are 

 several, the motions of which take place in essentially like cir- 

 cumstances, but with different phases. If, now, at any time t 

 the infinitely small alteration of the ergal occurs which is ex- 

 pressed mathematically as U changing into U + //.V, we have for 

 each single point, instead of /jl(V—Y), to construct a quantity 

 of the form /^(V— V), in which V represents the value of the 

 second function corresponding to the time t. This quantity is 

 in general not =0, but has a positive or negative value, accord- 

 ing to the phase in which the point in question was at the time t. 

 But if we wish to form the mean value of the quantity /^(V— V) 

 for all the points, we have, instead of the individual values which 

 occur of V, to put the mean value V, and thereby obtain again 

 the expression /x(V— V), which is =0. 



7. From the preceding it follows that, on the suppositions 

 made, we can put SL in equation (19) in the place of SU, so that 

 the equation becomes 



SL = ■=■ $v 2 + mv 2 S log i (21) 



The expression on the right may be simplified by introducing h 



m — 



for the product tj v 2 } which represents the mean vis viva of the 



point. Thence comes 



8L=aA + 2A81ogt. ...... (22) 



By the help of this equation we can determine the mechanical 

 work which is done in the change from one stationary motion to 

 another, differing infinitely little from it, without perfectly 

 knowing the motions, since to take into account the mean vis 

 viva and the time of a revolution is sufficient. 



