under the Influence of the Electric Discharge, 415 



stance must be taken into account. A part of the very large 

 quantity of electricity passing in each discharge charges the 

 surface of the tubes and discharges itself only very gradually 

 (in ten or more minutes), as is clearly shown by the afterglow 

 of the tubes. This is particularly striking when very long 

 tube conductions (30 metres and more in length) are employed. 



In considering galvanic batteries, such as Hittorf employed 

 in his beautiful investigations, we must first examine whether, 

 with the large quantities of electricity produced, the gas 

 returns after each discharge to its original condition. 



In order not to complicate the inquiry unnecessarily, let us 

 imagine the tube replaced by an indefinite space bounded by 

 two parallel planes. Let the whole of the gas be heated by the 

 instantaneous discharge to the same initial temperature A. 

 Let the distance of the walls be c, their temperature constant 

 and equal to zero, the temperature of the separate points be- 

 tween the plates u, their distances from the one plate x, and 

 the time t. Then the flow of heat, if a 2 is a constant, is deter- 

 mined by the partial differential equation 



du _ 2 d 2 u 

 dt~ a d^> 

 where 



u=f(x) = A when £ = 0, 



u — when x = and x=c, 



The solution gives, if r be a whole number, 



. 2r+l 



sin 7TX. 



(2r + l)7r o 



Let us consider only the temperature U in the middle of the 



c 

 plates, then x — x. If we expand the series and content our- 

 selves with the first term, we have 



U=4A— ^-. 



IT 



For a pressure p in air, 



a 0'25 . 760 centim. 

 p sec. 



according to Stefan*. 



If we take the pressure at T J n atmosphere, which corre- 



* Stefan, Wien. Ber. lxv. (2) p. 45 (1872). See also Kundt and War- 

 burg, Pogg. Ann. clvi. p. 194 (1875). 



