( 281 ) 



fi. In llic third pari of mv communications "< >n fourdimen6iona] 

 nets and (heir sections by spares", which is aboul to appear in these 

 "Proceedings" we shall find occasion t<> fix attention on the diagonal 

 planes presenting themselves in the sections of the (?,„. As any 

 vertex or rather any couple of opposite vertices <>t' C ia pos- 

 sesses an adjacent octahedron, the polygons situated in these diagonal 

 planes are always sections of octahedra. Probably the diagonal planes 

 presenting themselves in the sections of the C aoa were discovered 

 for the first time by Mrs. A. Boole-Stott, who made models of 

 these sections, and explained as sections with diagonal spaces by 

 Mr. II. W. Curjel. M 



The object of this paper is to study more closely the ca>es ; n 

 which the intersecting space contains one or more edges of (\,, and 

 C R00 : of the results revealed by these considerations these about (',„„ 

 have especially roused our interest. 



A. The spatial sections through an edge of C l6 . 



7. We consider the case in which the intersecting space contains 

 the edge AB of C la and indicate by A' and B' the vertices opposite 

 to A and B. Then all the vertices except A and A' are adjacent 

 to A and A' ', all the vertices except B and B' -<iw adjacent to B 

 and B' , and so the four other vertices 1\ , /\ , P 3 , l\ (tig. 2) are 

 adjacent to .1 and B at the same time. In other words: the octa- 

 hedra adjacent to A and /J, situated in different spaces, penetrate 

 one another in the square I\I\I\1\, the vertices of which they 

 have in common. So through the edge AB pass two diagonal spaces. 

 one of which corresponds to the opposite vertices l\, I\, the other 

 to the opposite vertices l\ , / J 4 ; they intersect the plane of the 

 square l\l\l\l\, perfectly normal in O to the plane through AB and 

 A' B' , respectively in the diagonals l\J\, l\l\- If / > s the trace 

 of the intersecting space through AB on the plane l\I\I\l\ , and 

 this line /, determining with AB that space, meets the diagonals 

 l\l\,l\l\ in the points S lt ,S s4 situated within the square, then 

 the section will show the particularity that the planes ABS lt and 

 ABS i4 are diagonal planes: so in some cases the edge AB will lie in 

 two diagonal planes. 



In the third communication "On fourdimensional nets, etc." quoted 

 above will be indicated that the particularity of an edge being situated 



] ) A series of these models, showing e.g. the decomposition of thé 120 ver- 

 tices el' tin' Cfloo into tin' vertices of five cells <\ :i , lias been inserted lately into 

 the collection of mathematical models of the University of Groningen. 



