of the Mechanical Theory of Heat. 173 



moveable point, expresses the theorem of the least effect. It is 

 true that in the signification there is a difference, inasmuch as 

 in deducing our equation we have supposed that the original and 

 the altered motion take place in closed paths which need not co-* 

 incide in any point, while in the theory of the least effect it is 

 supposed that both motions begin from a common point and end 

 at a common point ; yet this difference is immaterial to the proof, 

 because equation (24) can be deduced equally on both supposi- 

 tions, if in the one case we understand by i the time of a revo- 

 lution, and in the other that time which the moving point 

 requires in order to pass from the initial to the final position. 



Returning now to our more general result, expressed by equa- 

 tion (23), on comparing it with the theorem of the least effect, 

 its applicability is seen to be more extended, inasmuch as it in- 

 cludes also the cases in which the vis viva is altered by a transient 

 extraneous influence, or into which a change in the ergal enters, 

 whereas such cases are excluded from the theorem of the least 

 effect*. 



10. Having treated the simple case of a single point moving 

 in a closed path, let us now pass to more complicated ones. 



We will assume that there are a very great number of material 

 points which, on the one hand, exercise forces upon each other, 

 and, on the other, are affected by forces from without. Under 

 the influence of all these forces the points shall move in a sta- 



* It may, in passing, be further remarked that where the forces present 

 consist of central forces proportional to a definite (positive or negative) 

 power of the distance, the equations here developed are capable of being 

 combined very simply with the equation which expresses the theorem 

 of the virial. That is to say, in such cases the virial differs from the mean 

 value of the ergal only by a constant factor ; for when a force denoted ge- 

 nerally by (p{r) is determined by the equation 



<£(r)=£r w , 



in which k and n are constants, we obtain by integration, if we suppose the 

 arbitrary constant equal to 0, 



JV 



(r)dr=-A-r»+i 



and accordingly the equation 



^'(r)= n -±l^(r)dr 



is valid ; and hence the virial is equal to the mean value of the ergal mul- 

 tiplied by the factor -X-. Consequently the theorem of the virial can for 

 such cases be expressed thus : — The mean vis viva is equal to the mean ergal 

 multiplied by n ' r . It is obvious how all equations which contain the 



mean vis viva and the mean ergal can be simplified by the application of 

 this theorem. 



