of the Mechanical Theory of Heat. 1 77 



there results : — 



_2m(5-& lf +^Sy+Js^)=Sf8W + S^Slogi.(29) 



In this equation we now replace the products m-rw, m J&> 



m-p by the force-components X, Y, Z, whereby it is changed 

 into 



-^{XBx + YSy+ZBz=l l ^-8{v^)+Xmv^hgi. . (30) 



The left-hand side of the equation, thus transformed, must be 

 subjected to a closer consideration. 



12. Since, according to hypothesis, the forces operating in the 

 system have an ergal, in all cases in which, on the transition 

 from one stationary motion to the other, the ergal undergoes 

 only the change conditioned by the altered position of the points, 

 the left-hand side of the preceding equation is simply the varia- 

 tion of the ergal, and, as such, represents the work done in the 

 transition from one motion to the other, which we have denoted 

 by SL. "When, on the contrary, the ergal undergoes a further 

 change, containing a quantity which, as above said, is constant 

 in each stationary motion, but the value of which changes in the 

 transition from one motion to the other, the special circumstances 

 under which this happens must also be taken into consideration. 



For a single moving material point, it follows from our pre- 

 vious considerations that the work Sit depends on the phase in 

 which the point is at the moment when the alteration of the ergal 

 occurs. On the other hand, we have also further seen that with 

 a great number of points which are in different phases, so that 

 at the moment of the alteration of the ergal all the phases are 

 simultaneously represented, that difference vanishes for the mean 

 value relative to all the points, and that hence we may, as far as 

 the mean value is concerned, regard as the expression of the 

 work SL that variation of the ergal which the mere change of 

 position of the points supposes. 



Such a case is our present one, where in every kind of motion 

 occurring we have to do with very many points in the most 

 various phases ; hence we can replace the left-hand side of the 

 above equation simply by SL, whereby we obtain 



8L = 2,^8(v 2 )+Zmv 2 8\ogi. ... (31) 



13. In the preceding derivation it was specially presupposed 

 that all the points describe closed paths. We will now drop this 



Phil. Mag. S. 4. Vol. 42. No. 279. Sept. 1871. N 



