viii, a, i Gibbs, Williams, Galajikian: Methyl Salicylate IV 15 



sum will represent the total result of" hydrolysis at any time t. 

 This may be obtained experimentally by titration, and is the 

 quantity which we have represented by the symbol x. The fact 

 that x represents di-sodium salicylate as well as mono-sodium 

 salicylate is of no moment in this connection since both are prod- 

 ucts of saponification. Then, if x=xi-fx 2 , ^ =% 1 + ^r. 



dt dt dt 



Adding (9) and (11), 



^=KeiH-5Km2 (12). 



Substituting in (12) the value 0.005 for e and the value of n 

 obtained in (8), we have 



dx = °^°|K[y x 2_o.()944x+0. 018824— x— 0. 0428] 



+ ~ [t/x 2 — 0.0944x+0. 018824— x— 0.0428]- 2 (13). 



Let K 2 =^K and K = 5J. Then dx = 

 6 36 dt 



K 2 [/ x 2 — 0. 0944x+0. 018824— x— 0. 0428] + 



K 3 [l/x 2 — 0. 0944x+0. 018824— x— 0. 0428] 2 . 16 (I). 



The integral of this differential equation is a very complicated 

 expression containing the two constants in several different 

 forms, and it would be extremely difficult to determine them 

 from the integral expression after substitution of the experi- 

 mental values for t and x in Table V. It is possible, however, 

 to get approximate values for K 2 and K 3 by the application of 

 Simpson's rule to the differential equation. The degree of ap- 

 proximation can afterwards be tested, and the values of the 

 constants modified, if necessary. 



In applying Simpson's rule in this case, the method of pro- 

 cedure is as follows : 



Putting the equation into the form of a definite integral 



.f-f 



* x * dx (II). 



7 tj ™~ J X! K 2 [Z]+K 3 [Z] 2 , 



we see that the time t is the area under the curve plotted between 



values of x as abscissas and the values of T , r „-. T2 . rvi2 



tt-2 L^J+J^s L^J 

 as ordinates. 



" In the following equations, the term inside the brackets will be expressed 

 by Z for the sake of brevity. 



