( 84 ) 



polarimetric and copper tests both before and after inversion. Next 

 100 grammes of this glycerin were mixed with a 3 % starch 

 solution, warmed to 40° C. and kept at that temperature for a couple 

 of hours. After that the dissolved starch and dextrin was precipitated 

 with alcohol, filtered, a pinch of calcium carbonate was added to the 

 filtrate to prevent inversion by the slightly acid reaction of the 

 liltrate, and the alcohol was evaporated off. The syrupy residue was 

 dissolved in water, diluted to the volume of 100 cM. s and used for 

 the determination of the sugars by the polarimeter and Fehlings 

 solution before and after inversion. 



The original glycerin solution had contained 0.1 7% of glucose both 

 before and after inversion, while after the treatment with starch 

 100 grammes of the solution contained 0.60 grammes of reducing 

 sugars before inversion and 0.67 after that operation, which shows 

 that 0.43 grammes of glucose and 0.07 grammes of sucrose (?) have 

 been formed from the starch by the ferment. The polarisation of 

 the solution was -j- 0.9 before and -f" ^-4 a ^ tei ' inversion, giving 

 evidence, that notwithstanding the precipitation with alcohol, a small 

 amount of starch or dextrin has still remained dissolved. 



At any rate from the fact that the exclusion of oxygen prevents 

 the saccharification of the starch in the fruit and from the negative 

 results of the experiments on formation of sucrose by means of fresh 

 juice and of the precipitated and re-dissolved ferments, it follows that 

 the rapid transformation of starch into sucrose during the after-ripening 

 of some fruits is a vital process and not a consequence of the action 

 of some ferment contained in the fruit which, just as diastase forms 

 maltose from starch, could be isolated to form large quantities of 

 sucrose from any kind of starch in the laboratory. 



Mathematics. — "Congruences of hoisted curves in connection with 

 a cubic transformation" By Prof. Jan de Vries. 



(Communicated in the meeting of May 30, 1908). 



§ 1. If j\, a\, x 3 ,x i are the coordinates of a point A' with respect 

 to a tetrahedron having 1} O a , O z , 4 as vertices, then 



X l X 1 — X 1 x s — ' V S X 3 — X 4 X 4 



determines a cubic transformation which transforms the right line 



x] c = Xajc -\- fib i, 

 into a twisted curve tu :i , represented by 



1 



p.r'k = 



7.11k -f- fibk 

 The congruence r of the curves or' through the five points 



