BY A SPACE ROTATING ABOUT A PLANE. 7 



Resuming we can say that for the case under consideration the 

 projection on the plane t' (fig. 1) provides us with the means to 

 construct the rhombohedrical section, O A being the length of its 

 semiaxis, whilst AC and 6 on. represent the diagonals of the 

 limiting lozenges. We apply this in fig. 3 , where the rectangle 

 PP'Q'Q with the two lines P 1 Q l , B 2 Q 2 dividing it into three 

 equal triangles is repeated in an altered position, to the two 

 extreme cases of this first group of sections, i. e. for <p = and 

 for <p = 30°. By means of the rectangular triangles A 1 B 1 'C l and 

 A l B 2 'C 2 the axial diagonals J 1 C 1 and A 2 C 2 are found and on 

 these the lozenges A 1 B l C\B 1 and A 2 B 2 C 2 D 2 with transverse diago- 

 nals B l JD 1 = B 2 D 2 = 6 cm. are constructed. In accordance with 

 the fact that the section is a cube for <$ = , A X B X C\ B l is a 

 square. In all other cases the axial diagonal AC is longer than the 

 transverse diagonal BD; so, if we like, we may call AC the macro- 

 diagonal and BD the brachidiagonal. 



Now whilst <p increases from 0° to 30° the section (fig. 2) 

 changes in a simple manner. Whilst the intersection of the rhoni- 

 bohedron with the central plane % normal to the axis is and 

 remains the regular hexagon // G , the two endpoints A, A' of the 

 axis move away from 0, starting from a distance \\/ § cm., to 

 a distance 3 }/2 cm. So we may say that the cube corresponding 

 to the original position of the rotating space is stretched out in 

 the direction of one of its diagonals into a rhombohedron. 



If we suppose for a moment that the considered eightcell forms 

 part of an infinite pile of eightcells built up in the direction of 

 the edge PQ, so that the lower cube of an upper one coincides 

 with the upper cube of the next lower one, and we disregard the 

 limiting cubes common to two adjoining eightcells — as if we 

 wished to change a range of cells into a long vessel by removing 

 the interior diaphragms — we obtain a fourdimensional prism , 

 the right section of which is a cube. This prism will be cut by 

 any space in a rhombohedron , which only transforms itself into a 

 hexagonal prism with // 6 as right section for (J) = 90°; this result 

 will be applied directly. 



5. The case 30° < <$ < 00°. In this and in the following 

 case the rhombohedrical section of the space S.^%) with the 

 fourdimensional prism found above is truncated by two planes 

 perpendicular to the axis A A' in the points W , W' where that 

 axis meets the projections PP' and QQ' of the upper and lower 

 cubes. Here the points W, W' lie on PP 1 , Q 2 Q' ; so the end- 

 planes of the truncated rhombohedron bear equilateral triangles and 



