BY A SPACE ROTATING ABOUT A PLANE. 2 1 



extension of' the shading of the chessboard squares. So the shading 

 of the eighteells of the middle layer is shown in that of the cubes 

 (fîg. 6) forming their sections with the space S 3 (t) in its initial 

 position I, and in the sections of these cubes by the plane tt 

 (fig. 7); to obtain the shading of the eighteells of the plus and the 

 minus layer we have to invert black and white of the corresponding 

 eighteells, i. e. of the eighteells bearing the same number, of the 

 middle layer. Then two eighteells in space contact or in line 

 contact are alike, two eighteells in plane contact or in point contact 

 are different with regard to the shading. By adding that we suppose 

 the central eightcell to be white the shading is quite determined 

 by the stated rule of contacts. 



But there is more. For we can prove easily by this rule that 

 any two eighteells the projections of which on the plane x' coincide 

 are alike with regard to the shading. So the three eighteells of 

 G 3 projecting themselves (fig. 8) on the rectangle equal to P Q PQQ 

 and overlapping this for two thirds must coincide in shade, as 

 they are in space contact with the unique eightcell of G ] that 

 projects itself on P Q PQQ and therefore differ in shade from that 

 eightcell; the cube common to this eightcell and any of the three 

 belonging to C 3 projects itself on the overlapped part of P Q PQQ . 

 So we can go on and assert that the six eighteells of G Q must 

 correspond in shade, each of them being in space contact with at 

 least one of the three eighteells of G s and therefore differing in 

 shade with these, etc. 1 ) So we find easily the following result: 



White are the 



G x 



+ 



G, 



G, 



G c> 



+ 



Gq' + <'',' 



G* 



g q ' - g; 



4 1 eighteells . . . 



— 







black are the 

 40 eiu-htcells . . . 



• 



t G, 

 G x 



~G, 





+ G, 

 G Q 



6r™ 





, + G 3 ' 

 G C) ' G( 



- g; 



Now the question arises: "from the fact that the eighteells of 

 the same group correspond in shade and their sections are equal 

 the possibility of all equal pieces corresponding in shade presents 

 itself; in which of the cases I, II, ... VII is this possibility 

 realised?" To answer it we have only to consult the general table 



') We can give an analogue to the "rule of' contact" for the projections on sr' in 

 the form: "Two eighteells the projections of which overlap for one third or are in 

 point contact are alike, two eighteells the projection of which overlap for two thirds 

 or are in line contact are different with regard to the shading." 



