324 Prof. R. Clausius on the Application of a 



For the latter we will first introduce the sign of any function, 

 putting 



u=/(\). . ... ^ .... (5) 



The other function also, represented by mv 2 , can then be im- 

 mediately specified; namely, if/'(X) signifies the first derivative 

 from /(A,), it is 



8U=/'(X)SX. 



If we introduce this product into equation (5), and at the same 

 time replace the variation of the logarithm by its value, we obtain 



whence follows 

 or, dividing by 2, 



f'(\)8k=mv*^, 



!^=iv(\) (^ 



Further, from (4) we will form the following equation :- 





Vv 2 



Putting here for v 2 the value which results from the preceding 

 equation, we have 



V fiw 



/w ...... (8) 



Finally, we will represent yet a fourth quantity by X. Ac- 

 cording to the theorem of the equivalence of vis viva and mecha- 

 nical work, we have the equation 



in which E is a quantity which is constant during the motion, 

 and which we will call the energy. If the sum of the two vari- 

 ables on the left-hand side of this equation has a constant value 

 during the entire motion, the sum of their mean values has the 

 same value, and hence we can write 



E = U+|-^. 



By inserting here the expressions from (6) and (7), we obtain 



B=/(X) + i\/'(X) (9) 



Consequently, as soon as the form of the function /(X) is known, 

 we can, by virtue of equations (6), (7), (8), and (9), express by 



