22 ON THE SECTIONS OF A BLOCK OF EIGHTCELLS 



of results. We find that the answer is alternately no, yes, no, etc. 

 So in the cases II, IV, VI all equal pieces correspond in shade. 

 And of the other cases the first behaves differently from III, V, 

 VII, in this sense that in the cases III, V, VII all equal pieces 

 will correspond in shade, when the shading of the eightcells of 

 both the plus layer and the minus layer have been inverted, an 

 alteration which does not affect the first case. 



18. If in the solution of the problem of reconstruction of the 

 section of the block by putting together the different pieces, we 

 confine ourselves to the pieces belonging to the groups of the 

 middle layer, the result is a part of the whole section, included 

 between two planes. These planes are evidently the planes of inter- 

 section of the intersecting space /S 3 {t) with the two spaces sepa- 

 rating the middle layer from the plus layer and the minus layer; 

 as the two separating spaces are parallel to and equidistant from 

 the space containing the centres of the eightcells of the middle 

 layer, the two new planes, which we call -j- x and — x, are 

 parallel to and equidistant from x. In the initial position I the 

 intersecting space S 3 (x) coincides with the space bisecting the 

 distance between the separating spaces ; for this reason the planes 

 -j- x and — x are at infinity in that case and cannot appear in 

 fig. 6. For which value of the angle of rotation <J>, supposed to 

 increase from 0° to 90° do these planes -j- x, — x begin to inter- 

 sect the section? A single glance at fig. 8 shows us that this angle 

 P OA is characterized by its tangent ^ }/3 , so it amounts to 10° 53' 

 36"; in the rhombohedrical section of the block corresponding to 

 this angle the planes -)- x , — x would make their appearance in 

 the extremities A , A' of the axis of the rhombohedron. For any 

 acute angle that surpasses this value the planes -f- x and — x divide 

 the section into three parts, the extreme ones of which — equal 

 to one another — are left uncoloured , while the middle slice is 

 coloured yellow. On the side-faces of the sections the polygons of 

 intersection with the planes -\-ir, — x separate parts of the sur- 

 face differing in shade. What appears in the planes -\- x , — x 

 themselves is shown in the case II for both the planes, in the 

 other cases for the plane -j- x only ; in these sections can be seen 

 how the polygons are built up of the d-fnees and the d' faces of 

 the pieces of the middle layer. The other plane sections, added to 

 the figure-groups 10 m , 10 lv , . . . 10 vn , refer to the endplane not 

 visible in the principal figure. 



Alter what has been said it will not be necessary farther to 

 explain the meaning of the numbers without sign or with one of 



