648 report— 1884. 



D 8 A = - 2 A, D 16 B = §A a - |A'S, 

 D'-'C = n\A 3 T + |A 3 -^SA, 



Those expressions have to be introduced in Brioschi's disciiminant, and so will 

 give immediately the tactinvariant of p '- and a 3 for which we have been looking. 



6. On the ' Analysis Situs ' of Threedimensional Spices. 

 By Professor Walther Dyck, D.Ph. 



The following considerations refer to the analysis situs of threedimensional 

 spaces, and its formulation^ is called forth by certain researches on the theory of 

 functions, which, however, Tca'not enter upon here. 



The object is to determine certain characteristical numbers for closed three- 

 dimensional spaces, analogous to those introduced by Riemann in the theory of 

 his surfaces, so that their indentity shows the possibility of its ' one-one geometri- 

 cal correspondence.' 



Supposing every part of the space in question behaves itself as our ordinary 

 euclidian space, with this restriction, that the infinitely distant points are to be 

 considered as condensed at one single point (' Itaum der reciproken lladien '). 

 Collecting under one representative all those spaces, between which a one-one cor- 

 respondence is possible, we can form all possible closed threedimensional spaces by 

 the following procedure : 



We cut out of our space 2k parts, limited by closed surfaces, each pair being 

 respectively of the deficiency (Geschlecht) p v p w . . . p . Then, by establishing 

 a mutual one-one correspondence between every two surfaces, we close the space 

 thus obtained. The numbers p[,p„ . . . p of' the surfaces, thus made use of, and 

 the manner of their mutual correspondence then form what we may call the dis- 

 tinctive characteristic of our sp"aCe. This characteristic is determined : — 



1. By the existence of certaiu closed surfaces, which are not able to isolate a part 

 of the space. These are surfaces surrounding the above-named surfaces of the defi- 

 ciency p l ,p s . . . p K . 



2. By the existence of certain closed curves in our space, which can neither be 

 transformed into each other, nor be drawn together into one point. 



We will now consider the last-named characteristic, which, so far as I know, ' 

 has not been elsewhere discussed. Let me explain it by an easy example, suitable 

 to show the general particularities. Suppose two rings (of the deficiency 1) cut 

 off of our ordinary space. According to the manner in which the one-one corre- 

 spondence of these two surfaces is defined, essentially differing spaces are formed. 

 First,_for example, we can make them correspond so that meridian curves fall on 

 meridian curves, and latitudinal curves into latitudinal curves. Then there exist 

 curves which cannot be contracted into one point. For if we put a closed curve 

 surrounding the first ring, the curve, by all expansions and deformations it is liable 

 to, always encloses one of the two rings. On the contrary, supposing we had made 

 the meridional curves to correspond to the latitudinal ones and vice versa, curves 

 of the above description would not have been found. For a curve surrounding the 

 one ring can first be contracted into a meridional curve of this ring. This curve 

 is identical with a latitudinal curve of the second ring, and this last-mentioned 

 curve can be removed from the ring into our space, and therefore be contracted 

 into a point. 



In this way the particular correspondences, above described, between every two 

 of our surfaces giye rise to particular kinds of closed spaces. The enumeration of 

 these spaces is immediately connected with the enumeration of the canonical 

 orthogonal substitutions, which give rise to 2jo new periods from the periods 

 <Oj, <a 2 . . . <a 2P in the theory of Abelian integrals, according to Kronecker. 



1 hope to develope this subject further on another occasion. 



