( 487 ) 



"according to an - (J/J, where a according to circumstances can 



P 



"assume for even p one of the - values 1,2,...-, for odd p one 



"of the vahies 1, 2, . . . -— — ." 



2 2 



"For odd n the measure-poljtope Mp is intersected under the same 



L'a - 1 ^ 



"circumstances according to a (il/),) where a can assume for 



■P 



P P ^ P'\~^ 



"even p one of the - values i, 2,...-, for odd p one of the 



"values 1,2,. ..^^^." 

 2 



We sliall now, instead of losing ourselves in further generaHties, 



give the full results of the diagrams for the cases n = 4, 5, 6, 7, 8 



to make clear the above. In order to be able to indicate easily 



ratios of measure we shall suppose the edge of Mn to be unity of 



lengUi. 



3. Case ?z = 4. Tlie space — see first diagram — perpendicular 

 in the centre c of diagonal ae to this diagonal contains the six 

 vertices of M^ projecting themselves in c and cuts — see lines 3 

 and 4 — no edge ; so the section has six vertices. This same space 



cuts twelve faces — see line 7 — according to -(i/J and eight 



1 _ 



bounding bodies — see lines 9 and 10 — according to - {M^) ; so 



o 



the section lias twelve edges with a length |/2 and eight equilateral 

 triangles as faces. So the section is a (6, 12, 8) and, indeed, the 

 regular octahedron with edges \/2. 



Case ?z ^ 5. We find — see second diagram — thirty vertices 

 generated by intersection of edges, sixty edges, forty faces and ten 

 bounding bodies, so a (30, 60, 40, 10). The vertices are of the same 



kind, the edges have as - (JiJ the length - 1/2. The forty faces 



4 2 



consist of twenty - (J/g) and two times ten - {M^), i. e. of twenty 

 hexagons and twenty triangles, both regular ^) with sides — j/2. 



1) Where the regularity is obvious — as e. g. with the triangles by the 

 equal length of all edges, etc. — the additional "equilateral" or "regular" will 

 in future be left out. 



