( 283 ) 



So for (he position of the intersecting space adopted in fig. 3 four 

 diagonal planes ABS 1t ABS„ ABS„ ABS 4 pass through AB; the 



point of intersection S s of / and l\J\ tails on the production of 

 this side and does not lead -to a diagonal plane. 



On each side P 6 P t of (he starpentagon (fig. 3) there are remark- 

 able points besides the extremities P. , P 2 , which lead to faces and 

 no( (o diagonal planes, namely the points of intersection Q tf (2. 

 with the other sides and the midpoint M v If *S, coincides with 

 Q„ the sides PJ\ and l\l\ are cut in the same point and, the 

 two corresponding diagonal planes coinciding with one another in 

 the plane of intersection of the diagonal spaces ABP t P s and A BP 4 P^ 

 adjacent to P, and P t , this plane must contain the pentagon adjacent 

 to the edge P X P % of 6' 600 . So in this case the polygon situated in 

 the diagonal plane — compare in tig. 4 the planes normal to the plane 

 of the diagram in the lines qr and qr' - is a regular pentagon. 

 If aS\ coincides with M x the plane ABM — compare tig. 4 — , 

 being a plane of symmetry of the icosahedron, contains AB and the 

 edge parallel to AB. 



9. My second memoir with the title "Regelmassige Sclmitte u.s.w." 

 Regular sections and projections of C,, n and C eo0 , Verhandelingen 

 Amsterdam, first section, vol. IX, N°. 4, 1907 contains the data 

 that enable us to determine, for any position of the intersecting 

 space containing a certain number of edges of C 900 belonging to the 

 four groups of sections studied there, the number and the position 

 of the diagonal planes passing through any one of these edges, and 

 to construct the icosahedral sections situated in (hese planes. We 

 will try to explain this shortly. 



On the righthand side of the plates II, IV, VI, VIII has been 

 indicated how the icosahedra adjacent to the vertices of C 600 project 

 themselves on the axes OR , OF , OK Q , OB . In order to see at 

 a glance which sections normal to these axes do contain edges of 

 icosahedra — and therefore also of C e0Q - we consult the upper 

 lines of the plates XVIII, XVI, XIV, XII. We find then the 

 following table : 



^(12), ,,(12), ƒ,((>), 

 c,(3) , <',(3) , / 8 (6), A,(6) , i,(8) , 

 6,(5) , c,(10), /,(5), 0,(10), i,(10), 

 c 4 (30), ,,(60), 

 in which the indices 1, 2, 3, 4, distinguishing the groups, .correspond 



19 



Proceedings Koyal Acad. Amsterdam. Vol. XI. 



