Polarization at a Metallic Anode. 019 



that is. ( =- ) is an arbitrary function of the time, say 



F(t). v (/ ' rA =° 



By differentiating (4) with respect to x and making use 

 of the identity 



dc dc 



dx dt 



we obtain the differential equation 



^=D— .... (7) 



ami we require the solution of (7), subject to the conditions 



c = Ci throughout, when £<0, ... (8) 



-^=F(i), when x=Q, and *>0, . . . ( C J) 



c = c x at all times, when #=/, . . . (10) 



I being the length of the tube 



The solution of (7) under these conditions may be most 

 simply derived from the solution for the case in which F(£) 

 is a constant, say F. When 



(-£L= F ' ^ 



the concentration c at any point in the tube is given by the 

 expression 



c=d+(Z— j?)F— 2B W € x C0S T~ ' * ^ ' 



provided that 



1MTX 



2B m cos^p = (Z-#)F, .... (13) 



where the summation includes all values of m except m = ; 

 for (12) along with (13) is a solution of the differential 

 equation which inspection shows satisfies the conditions (8), 

 ( 10), and (llj. Equation ( L3) shows that (I— x)¥ may not 

 tpanded as an ordinary Fourier ►Scries in which in has 

 the values 0, 1, 2, &c, for the term in which m = is 

 excluded by the conditions. But it may be expressed as a 



!•! i i , 135 2n + 1 

 series ol cosines in which m has the values -, -. 7 ... — - } 



— — — z 



and which is equal to it for all value- of the variable including 



