Polarization at a Metallic Anode. 651 



is a solution which satisfies the conditions (8) and (10) and 



has the property that 



dc 



Thus 



— jT= 2 S*-F, when ,r = 0, and t>t r and <4*u-i. 

 «c r=0 



(-§)„.-*«> 



at all times, and (17) is consequently the required solution. 



For our purposes it may be considerably simplified. In 

 the first place we want only the concentration at the surface 

 of the electrode c z=0 , and putting ,c = in (17) we have 



Wl+ ii . w { i- ^1 / (2B+1 y }• (18) 



The second series in (18) converges very slowly and is 



impracticable for calculation unless I A D(t — t r ) is com- 



parable with unity. When I is (say) two or three centimetres 

 this i< not the case unless t — t r is very great (several days). 

 For all smaller times a tube a few centimetres long is equi- 

 valent to one of infinite length, so far as the effect on the 

 diffusion is concerned ; and we may replace (18) by the 

 limiting value to which it approaches when / becomes infinite. 

 To do this take the 1 in (18) inside the summation in n by 

 writing for it its identical value 



and (18) becomes 



8 °° 1 



7T 7(2^+1)"- ' 



8 7 £ 1 



c =0=01+ 2 67F.-72 , 9 , n , -. (19) 



When / = :c. the second series becomes now the sum of an 

 infinite number of infinitely small terms, and degenerates into 

 the definite integral 



c 



(2n+l) 2 

 Substituting: . . t 



and 

 tin- becomes 



2 



2 



1 f 1_*-* 



4 J. m J 



