( 491 ) 



The diagonal on which the intersecting space Spn-i is at right angles is 



one of tiie diagonals of the rectangle, e.g, FQ'. Tf the normal erected 



in the centre of PQ' on this line, representing the projection of 



the intersecting space Sj^n-i , cuts the side FP' in A, this point A 



1 

 always lies at a distance — , from the centre B of FF\ F'or 



•^ 2l/w — 1 



in the right-angled triangle AOF we tind that B is the foot of the 



norinal let down out of on ^ij and from this ensues ^i^.i^P= OjB" 



1 1 



and therefore AB=^~- -. — |/?z — 1. So A coincides for even n with 

 4 2 



the point of division Fn^ and this point lies for odd n in the middle 



2 



between Fn—\ and Fn-\-i . From this it is again evident that the 



2 2 



vertices of the section are vertices of J/„ for even n and centres of 

 edges of 3In for odd n. 



In the paper quoted above which restricts itself to the case n = 4: 

 we find in a note how we can regard the section under observation 

 as a "rhombotope" truncated at both sides; the course of thoughts is 

 as follows. Let us imagine in the direction of the edges FQ, F' Q' 

 on either side an infinite number of measure-polytopes Mn piled on 

 each other and let us then remove the measure-poljtopes Mn—\ , 

 projecting themselves on FF\ QQ' and lines parallel to these, with 

 which the successive poly topes i/„ bound each other ; then a prism is 

 formed with Mn—\ as right section. If this prism is intersected by 

 a space Sp^—i which projects itself along the perpendicular /„ let 

 down out of on FQ, the section is thus an Mr.—\ . What varia- 

 tion does this section Mn~\ of the prism undergo when we substitute 

 for the intersecting space projecting itself along l^ an other one 

 which projects itself along a line l^ through 0, enclosing with 4 an 

 angle </? As is easy to see from the figure this variation consists 

 of a, regular enlargement of the perpendiculars let down out of the 

 boundary of J/„_i on the space Spn—2 , projecting itself in 0, which 

 enlargement means a multiplication of those perpendiculars by sec <p 

 and can be regarded as a stretching in the direction of the diagonal 

 CD. As for n := -A, where Mn—\ becomes a cube, such a stretching 

 makes a rhombohedron of a cubs, out of i7„— i is formed in gene^-al 

 what we call a rhombotope. 



Just as the rhombohedron regarded as a whole passes into itself 

 when it is revolved 120^ about the axis, or - — in other words — 

 iust as the axis of the rhombohedron has a period three, the axis 

 of the rhombotope under consideration has a period n — 1. Let us 



