( 788 ) 
remains toxical. The presence of NaCl-solution is therefore necessary, 
though a very small quantity (some tenths of 1 eM° to the 5cM? 
serum) appears to be sufficient. Some titrations of diphtheria-serum 
treated thus, taught me already that the quantity of antitoxines 
decreases thereby only to a very low percentage. (Conversely, barium- 
sulphate, which as a dry powder leaves the anaphylactic powder 
intact, is able, in this same form, to fix the antitoxines). So where 
the serum-principle, toxic to hypersensible individuals, and the anti- 
toxical power of antidiptheria-serum can be dissociated according to 
this process, it seems to me that a priori a practical application 
thereof is not impossible. It is a purely technical probiem to work 
out these data more closely. 
About the reactions of immunity, which are enacted in the hyper- 
sensible organism, I hope to be able to give further information on 
a following occasion. 
Mathematics. — “Continuous one-one transformations of surfaces 
in themselves.’ By Mr. L. E. J. Brouwer. (Communicated 
Prof. D. J. Korteweg). 
(Communicated in the meeting of February 27, 1909). 
We shall treat in this paper an arbitrary surface, which in the 
sense of analysis situs is equivalent to a sphere, i. 0. w. which isa 
continuous one-one image of a sphere. We shall submit that surface 
to an enurely arbitrary continuous one-one transformation in itself 
and, we shall investigate whether this is possible without at least 
one point remaining in its place. 
To simplify we shall consider a continuous one-one image of the 
surface in a Cartesian plane; the infinity of it then of course forms 
an exception, because there the continuity of the correspondence 
is disturbed, and because it must answer as a whole to a single 
point; for that point we choose a point not remaining invariant in 
the transformation. 
In the Cartesian plane we indicate the points of the untransformed 
figure by letters without a dash, the corresponding point of the 
transformed figure by equal letters with a dash. In particular we 
shall indicate infinity by H and 4K’. 
We now construct in the untransformed figure the system of the 
circles around A as their centre. These closed curves & lie outside 
each other, expand from A to H and to each of those curves there 
