166 Prof. R. Clausius on the Second Axiom 



'dx ^ \ ., fdx 



in which ( -j- Bx J and ( ^- Bx j signify the initial and the final 



value of -j- 8,27. 

 eft 



As with an entire revolution the final is equal to the initial 



value, the equation passes into 



f £® & )^=° ( 10 > 



As to the terms on the right-hand side, it is first to be re- 

 marked that in the integration according to <£ the magnitudes 

 i and Bi are to be regarded as constant. Further, when any 

 magnitude dependent on <£ is to be integrated from to 1 (for 

 example, x), the following equation can be formed : — 



C 1 1 C 1 



1 xd(f)= - 1 xdt. 



Distinguishing the mean value from the variable quantity by 

 putting a horizontal stroke over the sign which represents the 

 variable, we can write 



xd<^ — x (11) 



I 



What is here said of the quantity x holds good also of the quan- 

 tities -p^Bx, \-rr) j and Bl-r- j , occurring on the right-hand 



side of the above equation. In reference to the last quantity it is 

 further to be remarked that the mean value of a variation is equal 

 to the variation of the mean value — that thus we can write 



<SM 



»" m 



Accordingly the equation obtained by integrating equation (9) 

 is the following, 



or, dividing by i, and transposing the first term on the right- 

 hand side to the left, 



Precisely similar equations to those here derived for the x co- 



