117 
as 9, measurable’), and we consider the pairs of points consisting 
of a point of 9, and a point of e,. A part of R&R determined by an 
element of g, and an element of 9, we shall call a paralleloelement. 
It appears as a continuous one-one image of a (h,n —h—1)-simplotope*). 
Let us call a division of a simplotope a into simplexes with one 
common vertex inside 2, whilst the remaining vertices lie in the 
boundary of a, a “canonic division”, then we can bring about such 
a canonic division by first executing it for the two-dimensional limits, 
then for each three-dimensional limit by projecting the divisions of 
‘its two-dimensional limits out of an arbitrary inner point, then for 
each four-dimensional limit by projecting the divisions of its three- 
dimensional limits out of an arbitrary inner point, and so on. 
Accordingly we can divide the paralleloelements of R into (n—1)- 
dimensional elements in such a way, that by their mode of being 
joined they cause A to appear as a closed (n—1)-dimensional space. 
Let p be a paralleloelement of &, d, resp. d, the corresponding 
element of 0, resp. 0,, 4,A’,....A,“ a positive indicatrix of d,, 
A,A,....A,—'-)) a positive indicatrix of d,, we then define the 
row of pairs of points (4,A,), (4',4,),....(4,A,), CA AD oasis 
(ADA, HD) as a positive indicatrix of the partitional simplex of 
p determined by those pairs of points, and with the aid of it. we 
fix the positive indicatrix of the elements of Z belonging to p. In 
this way we determine of all elements of A the positive indicatrix, 
where for two arbitrary elements having an (n—2)-dimensional limit 
in common these indicatrices satisfy the relation prescribed for 
tivo-sided spaces. 
So Mè is a closed two-sided (n—A)-dimensional space. 
The set of the vector directions of Sp, forms likewise a closed 
two-sided (n—1)-dimensional space (of the connection of the (n—1)- 
dimensional sphere) which. we shall represent by B. The positive 
indicatrix of the spheres of Sp, (and with it at the same time the 
positive indicatrix of B) we determine by regarding them as boundary 
of their inner domain. 
If we conjugate to each pair of points consisting of a point of 9, 
and a point of 9, the direction of the vector connecting the two 
1) ibid, p. 98—100. 
2) Let in Sp,»—1 be given a plane h-dimensional space v and a plane (n—h—1)- 
dimensional space w. Let S, be a simplex in v, Sw» a simplex in w. The set of 
those points of B, which in the direction of w project themselves on v in S, 
and in the direction of v project themselves on w in Sw», form by definition an 
(h, n—h—1)-simplotope. Of a simplotope the limiting spaces of any number of 
dimensions are likewise simplotopes. (Comp. P. H. Scrourr, “Mehrdimensionale 
Geometrie”, Vol. I, p. 45). 
