( 543 ) 
Mathematics. — “On fourdimensional nets and their sections by 
spaces.” (Fourth part). By Prof. P. H. Scour. 
The net (C,,). 
1. In the first communication under this title we have transformed 
the net of the cells C@ into a net C,,,) in two different ways, into 
4 J 
a net of cells C@ and into a net of cells Ge? The difference 
between these two transformations may be characterized by the 
remark, that each cell Cf contains as a part the cell Cc? from 
which it is derived, whilst it is possible to consider each cell OM? 
to be bodily inscribed in a cell Oke by starting from two nets of 
Ge each of which fills the space Sp, entirely, related to one another 
in such a way, that the system of the vertices of the cells of the 
one is at the same time the system of the centres of the cells of 
the other and reversely. As we have used the second of these trans- 
formations in the deduction of the table of relations between the 
axes inserted in the first paper, we still cling to it here, though it 
cannot be denied that the advantage of including the cells Cie in 
boxes C§” is not quite so important as was that of including the 
cells oo in cells Ce, Cc! all cells (ee of the net corresponding 
with one another in orientation. 
We again restrict ourselves to the sections of the net (C,,) by 
spaces normal to one of the four different kind of axes of one of 
the cells cy and therefore of all the cells of the net. We remember 
to that end that the table on page 544 quoted above indicates which 
diameters of the box CY correspond to the chosen axes OF,,, OK. 
24? 
OF}, ‘Oli, ot CS. We repeat here the part of it relating to the 
net (C,,) in the form 
(4) (8) (3) (6.7 (5) (1,2) 
Peo, Okn(2lT 0)Cs, OF y=OKe=(3,1, | Cs, ORa=OR OFS, 
indicating by means of. the figures (4), (8),..., (5), (4,2) between 
brackets the lines of the table, where these results are to be found. 
By this it is immediately evident, that the series of sections normal 
to OLF,, and to OK,, involve every time a definite position of the 
