352 

 or 



r' = {i —e-ki'y{i -]- ^hy (U) 



From (10) and (11) follows: 



V (1 + i A)» 1 + A + I A^ + ^ A» ■ • • ^ "'' 



This fraction is always <il, also when A is negative, because 

 the third power term can never preponderate. The latter is easilj' 

 proved as follows.- 



If A <^ 0, it is required for the preponderance of the 3'*^ power 

 term that: 



- ^ A' > I A% hence A < - 9, 

 hence : 



g-fci / _ g-pk, < <^ _ 9 ^ 9g-A:i t^ 



or 



Se-h t 4 g-ph ' > 9, 



and this can never be fulfilled, because e~^'^' and e— /'^i' are always 

 smaller than 1. 



It follows therefore from what precedes that when p deviates 

 from the value 1, no matter whether p^l, or <^ 1, always less 

 glycerine will be liberated during the saponification than corresponds 

 with g= T\ 



§ 4. Let us now consider the esterification which proceeds ac- 

 cording to the same scheme. Then we find : 



g = e-(p+'i)^^i (13) 



and 



T = }è-P'^t -^ ^e-^f (14) 



We now put the question : what is for values of p > 1 the 



situation of the /{g, T) = determined by (13) and (14) with 

 regard' to the curve g =^ 7'^ 



We find from (13) and (14) in the same way as before: 



1 + — rsr— <'5) 



and 



1 + *^ 1 ■ • ■ • ('«) 



Let us put the fraction : 



g—pktt — g—ht 



