203 



/>„ h^ b, />,, . . . . />„_! b„ 



a„ ffj (f, .... «/,_2 «.'i-i (f„ = O 

 A displacement of tlie symbols /; each a i)lace to the right and 

 of the symbols a each a place to the left reestablishes the üi-st 

 condition. 



For the first element of the series we can now write the equation 



^'^0 + ^"i = ''^1 + Va^ 



or 



b„ + flxt^ — h^ 4- xa^ (3) 



If now we suppose that we are waitijig fii-st for a complete setting 

 in of the equilibrium before filtering-, it follows that r^^ = 6^ , rï^ = /^, 

 etc. so that the above equation becomes : 



b,^xh, = {\-yx)h^ 



and further: 



b, + xb, = {l+x)b. 



b„-2 H- xbn = (14 .V) />„_! 

 è„._, z={l-]-w)bn 



From these n equations we can eliminate ij, (62 ••• ^«-i after which 

 there remains a relation between h^, bn and :v which may be written 



b ^•n+l 1 



7^= ^^==1 +-^' +^''- + --. + -^'" = —• . . {A) 



On X i 



From this expression for U can be deduced : 



If n is constant, U increases indefinitely with x. 



If X is constant and > 1 : t/ increases indefinitely with n. 



If X is constant and <[ 1 : f/ increases with n, but is maximal 



If ^ is kept constant, x decreases with increase of ??, but is minim- 



1 



ally equal to 1 . 



Hence, in the most favourable conditions there must be added, 

 in order to lixiviate a mass with a volume of liquid v to a degree 



of lixiviation U, a volume of water > — -v. 



- K 



As a rule the expression found for (J cannot, however, be applied 

 to the methods of leaching used in practice. 



If we mix a mass, which in some non-chemical manner holds a 

 liquid in combination, with a more diluted solution, these solutions 

 will not mix immediately, because as a rule the same force that 

 causes the combination of the first liquid will oppose the immediate 



