of Iron and German Silver. 483 



assumed by the bar in any initial condition when its ends are 

 suddenly brought to the temperatures u^ and w,. Hence we 

 obtain the state of the bar at the end of the 0th period. We 

 have to solve the same problem for the 1st period, in which the 

 ends have the temperatures u^ and Wq, taking as initial state that 

 of the bar at the close of the 0th period. From this we lind the 

 state of the bar at the close of the 1st period, which forms the 

 initial state of the next following period. Proceeding in this 

 way we may easily satisfy ourselves that the distribution of tem- 

 peratures in the bar very soon approaches two limiting states, 

 which continually repeat themselves, and one of which belongs 

 to the even, the other to the odd periods. Both of these limiting 

 states are absolutely independent of the arbitrary distribution of 

 temperatures at the commencement of the 0th period. 



We will now give the result of these considerations, and for 

 this purpose introduce the following symbols : — Let 



X be the distance of any point in the bar, reckoned from that 

 end which in the 0th period possessed the temperature Wq, 



T the duration of a period, 



6 the time within the current period (^ = commences the 

 period; ^ = T closes it), 



Uq and Wj the alternate temperatures of the end surfaces of the 

 bar, 



Va^ and Vg^+i the temperatures of any point at the time 6 in 

 an even and in an odd period respectively, 



I the length of the bar, 



U the temperature of the surrounding air, 



K and H the internal and the external conducting-power, 



P and Q the circumference and the transverse section of the bar, 



C the specific heat, 



D the density. 



Lastly, let us put (for abbreviation) 



^=cd' ^""qcd' ^"^^ 



Then is 



V.,+, = U + (^'-u)(Z„ + Z,)+^(Z„-ZJ 



SttA , .^^ n . 2niT e-P""^ 



I ^ Pn ^ i + ^" ^" 



2 12 



(2) 



