10 Prof. R. Clausius on the Relations between the 



for X, 



m (dr\* . . m 3 1 /m 



ay^^'-f? (9) 



The second equation reads, in the form given under (5 c) : — 



8jU =. — Sv 2 + mv^B log t. 



In order to apply this equation, we have first to form the ex- 

 pression of the ergal U for the vibration-motion. In accord- 

 ance with equation (8), we may put 



U=j(F'(r)-m^l)rfr, 



from which by integration, writing F(r) for the integral of F(r), 

 we get 



m a 1 



V = Y(r)+~c 



2 r 2 



This is such a function as was mentioned above, containing a 

 quantity c which is constant during each motion, but may change 

 its value on passing from one motion to another. This quantity 

 must, when we form the variation ^,U, be regarded as constant ; 

 and thus we obtain 



8,U=«F(r) +5(«J. 



By introducing this expression into equation (5 c), at the same time 

 replacing v 2 by ( -j J and using instead of i the symbol chosen 

 for the period of a vibration, i J} we transform that equation into 



afW + f C 4=f<|)%K|)^io gi , • do) 



From the two equations (9) and (10) we can eliminate the 



Tdr\ 2 

 quantity ( -j J . We will, however, for the present, only employ 



/dr\ 2 

 in the variation of \-j-.) the expression given in (9). At the 



same time it is to be noticed that this is a complete variation, in 

 which the change of the quantity c must also be considered. 

 We thus get 



Isg^i^-fc^-m^. . (11) 



