490 UNIVERSITY OF VIRGINIA PUBLICATIONS 



method was that if the temperature changes be slow and steady, no percep- 

 tible mixing is caused by rise or fall of 0.5°, while experience showed that a 

 temperature change of only 0.1° caused by a break in the heating circuit 

 of the thermostat was always fatal to results. Of course, in the latter 

 case the fall was comparatively rapid. 



MATHEMATICS INVOLVED IN THE PROBLEM. 



Consider a substance diffusing in the direction indicated by the arrow in 

 figure 2. Let u be the concentration at A, and u—du the concentration, 

 at B. If now we make the thickness of the layer very small bu becomes , 

 du. Hence we have the expression. 



(u + du) — u 



for the difference in concentration at A and B. Differentiating this ex- 

 pression we obtain for the change in concentration corresponding to dx, 

 the thickness of the layer. 



du d^u du _ (fiu 

 bx da;2 bx bx" 



The change in concentration with respect to time is, — , hence we can 



write, 



,-.> bu , b^u 



® ■ br=^-^- 



Any solution of equation (I) must be a function of x and t. Put 



u = f {x, t) = cos nx • e~*"'', 

 differentiating with respect to t and we have, 



OU TO — knH 



— - = — k- n^ ■ COS nx ■ e , 

 bt 



differentiating with respect to x we have, 



