MORSE AND PIERCE. — SUPERSATURATION IN GELATINE. 631 



will go through a uuit cross-section in a unit of time under head of unit 

 gradient of concentration. Let a 2 (essentially positive) be the diffusion 

 constant of silver nitrate; then the quantity of silver nitrate that will 

 cross a section s of the tube at a distance x from the origin in a time 

 St is the diffusion constant X gradient X time X area of cross-section. 

 Therefore 



Q = - a 2 ^ . 8t . S (3) 



dx 



The quantity crossing a section at distance x -f 8x is 



9Q „ 9 9u . a .9 



a, 



Q + -^ 8x = - a 2 ~ 8t S - a 2 \^ . 8t . 8x . S (4) 



C/X C/X c/X 



The accumulation of silver nitrate in the region between x and x + 8x 

 is the difference of these quantities, 



a 2 ^—3 . 8t . 8x . S 



dx 2 



This accumulation may also be expressed as change of concentration 

 multiplied by the volume, = 8u . 8x . S ; 



9 2 u 

 . ' . a 2 =-3 8t . 8x . S = 8u . 8x . S ; 

 dx 2 



whence, dividing by 8t . 8x . S 



9 2 u 9u 



(i 



2 



(5) 



9x 2 9t 

 This is Pick's Law.* 



Analogous considerations give us an exactly similar equation for the 



concentration of the Cr0 4 ion at the point x and time t ; namely, 



h i &v 9v 



b W = 9t . (6 > 



where b 2 is the diffusion constant for potassium chromate. 



In deriving these equations we have made the assumption that the 

 formation of earlier precipitates does not materially affect the head 

 under which the ions accumulate for the formation of new precipitates. 

 If we make this assumption we can solve the problem completely. 



Equations (5) and (6) are exactly alike, but their solutions will be 

 different because they have to satisfy different initial conditions in the 

 two cases. The lower end of the tube was kept constantly, by occasional 



* Pogg. Ann., XCIV. 59 (1855). 



