BY A SPACE EOTATING ABOUT A PLANE. 23 



the signs + on the side-faces of the sections of the block, on the 

 endplanes of these sections and on the sections of these section's 

 with the planes -|- % , — %. With regard to the letters A, B, C, 

 D, E, indicating the sections of the separate eightcells of different 

 shape, inscribed also in the endplanes and the sections with the 

 planes ~\- ir , — tt it may suffice to remember that the forms A 

 and B meet the invariable plane tt respectively in regular hexagons 

 and equilateral triangles marked on them, whilst the other forms 

 C, JD, E have no point in common with that plane. 



19. A mere inspection of the groups of figures 10 n , 10 m ,. . ., 

 10 vn can show how the section of the block composed of the 

 sections A, B, C, . . . of the separate eightcells changes, when the 

 angle of rotation ty varies from 0° to 90°. from which ensues that 

 the angle vp of the side-faces of the section with the plane t 

 increases from 54° 44' 9' to 90°. If we now push the rotation 

 still farther and suppose that <p varies from 90° to 180° we pass 

 at the supplementary values of these considered above by stadia 

 VI', Y', . . . , I' closely connected to the sections VI, V,..., I. 

 For, if we consider the two positions (p and ISO — <P of the inter- 

 secting space S 3 (t) as each other's mirror-image with regard to 

 the initial position <p = of that space as a threediniensional 

 looking-glass, and remark that the reflection of the block itself in 

 that looking-glass interchanges only the signs -j- and ■ — of the 

 eightcells of the plus layer and the minus layer, then it is evident 

 that the two sections corresponding to two supplementary values 

 <P, 180° — cp of the angle of rotation are each other's looking-glass 

 image with regard to the plane ir , the interchange of the signs 

 + included. 



We finish our considerations by a rapid survey of the different 

 cases. 



Case II. The sections differ but slightly from the initial case 

 (fig. 6). If we invert the shading of the pieces corresponding to 

 the plus layer and the minus layer, and we glue together the 

 corresponding pieces (1 , -f- - 1), etc. of the groups G Q , -f- G & and 

 (3, — 3), etc. of the groups G e ', — G ( .', we hit upon a figure 

 which can be derived from the solid (fig. 0) by stretching in the 

 direction AA'. As we already remarked the shading of the equi- 

 lateral triangles in -|- t, — tt correspond to the r/-faces and the 

 öf-faces of the middle layer : if the shades were inverted they 

 would refer to the «/'-faces of the plus layer and the effaces of the 

 minus layer. 



As a glance at the figure shows, the section with the plane 4- k 



