( 387 ) 



why this particularity as was already stated there - does not 



present itself in any of the sections of the four principal groups. 



A mere inspection of plate 11 is sufficiënt to show, that all the 

 sections of C\ 8 represented there - not only those related to the 

 axes 0E lt , OK„ t 0F W 0R lt but also those of the last column 

 agree with one another in this, that any edge is situated in one and 

 only one diagonal plane, moving parallel to itself if the intersecting 

 space displaces itself parallelly. As an example we fix our attention 

 on the figures of the fourth column, where a hexagon MA US CD 

 starts on half its journey as a line MX to end it as a lozenge 



MAN C. 



Now the reason, why no edge situated in two diagonal planes 

 occurs here in the cases of sections by spaces containing edges of 

 C lt , can be derived from plate I. It comes to this, that spaces 

 through edges of C 16 , not leaving that cell entirely on one side, do 

 not present themselves for sections normal to OR ls or OF 19 , that thej 

 pass through the centre O for sections normal to 0E lt or 0K lt 

 ami contain a face of C lt for sections under the heading (3, 1,1, '1)0 K 9 . 

 If the intersecting space see fig. 2 of the communication of 



October — contains not only the edge AB but also the centre 

 of the cell, the two points of intersection S lt S %4 coincide in 0, and 

 instead of two diagonal planes ABS lt , ABS„ we find only one 

 diagonal plane ABO, containing also the edge A' IV opposite to AB 

 and therefore intersecting the section in a square; this happens in 

 the cases of the last figures of the first and the third column of 

 plate 11, for the first column with each, for the third column with 

 only one diagonal plane, represented horizontally. In the case of the 



4 

 last column corresponding to the fractional symbol — the triangle 



OPQ forming the base of the section is a face of C lt ; so through 



any side of this triangle passes only one diagonal plane. 



5. In order to determine the threedimensional space-fillings gene- 

 rated by intersection of the net (C lt ) we can follow different ways, 

 some of which are of a more theoretic, others of a more practical 

 character. Those of the first group correspond in this, that we deduce 

 from the section of a determined ( \, with the intersecting space 

 how this space must affect the other cells of the net (C 1§ ). So we 

 can project the axes of all the cells, normal to the intersecting 

 space on the axis taken as axis of projection, and deduce from the 

 fraction corresponding to the chosen C u the fraction corresponding 

 t0 ;U1N other cell Of the net: this method has been applied to the 



