^284 ) 



to lliose of the groups of icosahedra on (II 6 ), [V 6 , VI 6 , VIII 6 , whilst 

 the numbers placed between brackets indicate how many edges lie 

 in the intersecting spaces. However the cases a x , a„ a 9 can be left 

 out, as referring to intersecting spaces leaving the C eoo totally on 

 one side and being therefore unable to furnish sections containing 

 diagonal planes; for each of the sixteen remaining cases the trace / 

 of the intersecting space on the plane of the pentagon adjacent to 

 the chosen edge must be constructed. These traces, indicated by the 

 symbols d lf e x ,...,e 4 of the rases to which they belong-, are repre- 

 sented altogether in iig. 5. 



10. The determination of the trace / causes the least trouble if 

 this line contains two of the remarkable points /> , Qi, Mi corre- 

 sponding respectively to a vertex, a point of intersection of two 

 non-adjacent sides and the midpoint of a side of the starpentagon. 

 In order to divide the difficulties we treat these simple cases first. 



Case d x . On plate II ; ' we find under d that the groups I and VII, 

 each containing four icosahedra, furnish faces situated in the inter- 

 secting space, whilst group III gives six icosahedral sections through 

 two opposite edges. So the trace c?, to be found passes through a 

 vertex P, and a midpoint Mr, if /', is taken as 1\ ■, then JA f must 

 be either J/ s or M A . So we find that the trace d x coincides with 

 one out of ten homologous lines, if by "homologous" lines we mean 

 lines passing into one another cither by a rotation of the pentagon 

 about its centre .)/ to an amount of any multiple of 72° or by a 

 reflexion with respect to one of the lines M l\ as mirror, i.e. in 

 general by any transformation that transforms the pentagon into 

 itself. 



The line d l cuts two other sides, the sides f.P, and I\/\, of 

 the starpentagon ; as 1\ does not lead to a diagonal plane, any of 

 the 12 edges lying in the intersecting space is contained in three 

 diagonal planes. These new diagonal planes are connected with the 

 groups IV and VI, each of which contains 12 icosahedra. x\s the 

 section passes rather near the centre Mi in the case of IV and 

 rather near one of the extremities l\ in the case of VI, it is prob- 

 able that IV corresponds to the point on I\P^, VI to the point on 

 P 3 1\. Later on we will prove this to be true. 



We will add the remark, that the number 12 of the edges lying 

 in the intersecting space is given back by each of the groups I, III, 

 IV, VI, VII, the corresponding diagonal planes the /aces of land 



VII included - - containing successively 3, 2, 1, 1, 3 edges. 



Case <v On plate VI 6 under c the group 1„ leads to a point Q 



