the Elementary Law of Hydrodiffudoii. 493 



For x = l x + Z 2 = L, for all values of t 9 



fe= ' ^ 



since at the upper electrode no salt goes out, and at the lower 

 electrode no salt enters. If the moment of placing in layers 

 one over the other the two concentrations e 2 and z l be taken 

 for the initial point of time, the following hold as the initial 

 conditions of the experiment: — For £ = 0, 



z=z{ for all values of x from x = to x—l^ \ ,.<. 



z=z 2 ,, „ „ x—\ to x=l l + l 2 = L. j 



A particular integral of the differential equation (1) is 



z= (A cos hx + B sin hx)e' mt . 



The three constants A, B, and h are to be determined from 

 the initial and limiting conditions of the experiment. 

 In the first place, 



~ = ( — hA sin hx + AB cos hx)e~ hm 

 O x 



in the layer #=0 is to have the value for all values of t; 

 hence B must be put =0. Further, the same differential 

 quotient for x = h is likewise to vanish for all values of t; the 

 constant h must therefore be so chosen as to satisfy the equa- 

 tion 



JlL = W7T, 



where n = Q, 1, 2, 3 . . . 



The sum of the particular integral 



°° . (rm \ - n -^kt 



z = n"2A n cos[j-x)e ^ 



gives the general integral. The constant A , yet to be deter- 

 mined, can be ascertained from the initial conditions (4): — 

 For £=0, 



z=n,y,A„ cos ( t~ x ) =z i f° r a ^ values of x from x = to 

 o ^ L / x=l i; 



and 



z— 2 A ?t cos ( -y- x J = z 2 for all values of x from x = l x 

 o \ h / tox=l 1 + l 2 = L. 



According to Fourier's theorem we have 



