2G8 Prof. R. Clausius on a new Mechanical Theorem 

 4>i) $& ■ • • 4>n determined by the equations 



t = i l (f) l = ^(f >Q ... =i n <j> n 

 can be regarded as constant , satisfy the condition that the sum 



IpS^q — hSk 

 _ _ 



has a mean value that vanishes as the time increases, then the fol- 

 lowing equation is valid: — 



S(U-f) = ^81ogt + 2^8o ) . . . (II.) 



in which the first sum on the right-hand side, like the sum pre- 

 viously mentioned, contains n terms, which correspond to the n 

 variables q p q 2 , . . . q n j while the second sum ?°efers to the quanti- 

 ties c,, c 2 , fyc. contained in U, which are constant during each mo- 

 tion, but change their values on the transition from the one motion 

 to the other. 



Equation (II.) is my equation in the generalized form men- 

 tioned at the commencement. While in Hamilton's equation (I.) 



the integral 1 2Tdt is to be regarded as a function of the vari- 

 ables q x , q 2 , , . . q n , of their initial values k v A* 2 , t . . k n , and of the 

 energy E, and in equation (I, a) the integral I (T-XJ)dt as a 



function of q x , q 2 , . . . q n , k 1} k 2 , . . . k n , and t, in this equation the 

 mean value U—T appears as a function of the time-intervals 

 i v ? 2 , . . . i n and the quantities c v c 2 , &c. It also can be analyzed 

 into as many partial equations as there are independent varia- 

 tions on the right-hand side, whereby, however, we of course 

 obtain quite different equations from those resulting from the 

 analysis of Hamilton's equations. 



8. In order to demonstrate the theorem,, let us form for any 

 one of the n variables the product ph t q, and differentiate this 

 according to the time. Thereby we obtain 



=m 



*t% 



Into this, for the abbreviated symbol p, we introduce from (6) 



