113 
value — 4,08 already now, we see clearly that there is a region of 
unmixing at hand, and it will already have appeared with some 
0) dlp, 
decrease of temperature. For /= 0,85 finally Ee has already got the 
Ae : 
/ É dip, 5 
negative sign = = — 0,3] at the chosen temperature; so the 
av sf 
vapour-pressure line does not descend from the side of the most 
volatile component, but ascends; there is a maximum vapour pres- 
Up: 
sure. But then the value of 
a has fallen to almost —6, and we 
at 
may expect that even for not very great concentrations unmixing 
will take place. 
The calculations given here, will, I hope, have sufficiently eluci- 
dated the thesis which I pronounced in the cited paper that van ’r 
Horr’s rule need not hold, even as a limiting law, for systems whose 
components differ very much in vapour-pressure, when viz. these 
substances do not mix in all proportions, or at least a region of 
unmixing is close at hand. They also set forth again *), how much 
greater the influence is of slight deviations in the value of / from 
unity, than in those of bs or 2 and that such deviations are able to 
1 a, 
modify the course of phenomena entirely, so that certainly only a 
small part of all the possible cases is observed when we start from 
the supposition that the relation 7= 1 should be always rigorously 
fulfilled. On the other hand they also show that in all the systems 
known to us, we have to do with values of / which are contained 
within narrow limits, and that we have not a single indication to 
think values possible for the value of / of the same order as 
b a 
undoubtedly occur for a and also for —. 
a 
1 1 
Mathematics. — “On looping coefficients.” By Dr. L. K. J. Brouwer, 
(Communicated by Prof. D. J. KortTewuG.) 
(Communicated in the meeting of February 24, 1912). 
Let us suppose in Sp, two non-intersecting simple closed curves 
k, and k, furnished with a sense of circuit. Then #, possesses with 
respect to k, a looping coefficient answering to the intuitive notion 
1) Cf. the paper in the Zsch. f. phys. Ch. 75 cited in the beginning of this 
treatise. 
pe 5 
Proceedings Royal Acad. Amsterdam, Vol. XV. 
