270 Prof. R. Clausius on a new Mechanical Theorem 

 mark for mean values, we can put 



tj dc dc 



Meanwhile, on the left-hand side the integral-symbol may 

 remain, and only the variation-symbol 8 t be transposed, which is 

 admissible with variations where tis regarded as constant. Then 

 the equation is 



S ( gf t (U-T) ( »] = -S ^7 A8 * + sg8 e . . (16) 



Here, instead of variations in which the time is regarded as 

 constant, we will introduce on the right-hand side variations in 

 which the phases belonging to the variables in question are con- 

 sidered constant. 



The procedure to be employed in this alteration is easily found, 

 as follows. Let any quantity dependent on the time be signified 

 by D, we will put, with the original motion, 



Z = F(0, 

 and, with the different motion, 



Z* = F(**) + eF 1 (**), 



in which t and t* represent times corresponding to one another, 

 F and F, signify any two functions, and e is an infinitesimal 

 constant factor. If now the variation S t Z is to be taken, we 

 have simply to put t* = t and then to form the difference Z* — Z, 

 whereby we obtain 



while if the variation S$Z is to be taken, for we must put that 

 value of the time which corresponds to an unvaried value of <£, 

 namely 



and then again form the difference Z* — Z. We have thus 



S (p Z = F(/+V)+e'F 1 0+^/)-F<O. 

 This gives, neglecting terms of higher order with respect to 8<pt 

 and 6, 



S,Z = eF,(0 + ^V» 



which, according to the preceding, we can also write thus, 



8*Z = 8,Z + ZV. ..... (17) 



An equation of this form is to be constructed for each of the 

 variables q x , q q , . . . g n , wherein the phases <j> lf ^> 2 , . . . (f> n are to 

 be successively employed. We thus obtain for q v) with a little 



