484 M. Hermann Herwig on the Independence of Temperature 



we obtain 



g = -A(l+«*)(l+/3O + B(/-0)(l+«O. 



From this differential equation, only the first differential quo- 

 tient is, in the first place, derived, viz. 



— tit - — \- const. 



The integration-constant here occurring is determined from 

 the circumstances that for the end of the rod, where alone a cur- 

 rent-action is now to be perceived, a temperature t' no longer 

 variable prevails ; therefore 



For the further consideration the heat developed, in the unit 

 of time, in the entire rod is now calculated, and that given out 

 put equal to it. This leads to an equation w T hich is to be con- 

 ceived as an infinite sum of equations of the preceding kind. If 

 X is the length of the rod, and T the initial temperature, then 

 the equation is 



—0 



The value ( -j- \ is the previously determined differential quo- 



tient -j- for the temperature T, while the integration-constant 



of course possesses the value stated. The two other terms of 

 the present equation contain as a single complicated constituent 



the integral 1 tdX. This integral, however, is simply equal to 



the mean temperature of the entire rod multiplied by the length 

 of the latter. Naming the mean temperature m, the equation 

 therefore becomes 



=Bx{-*-!+ TO ( 1 -!/ 3 )}. 



