BY A srACE ROTATING ABOUT A PLANE. 



9 



how to find the images of the different sections in parallel projection , 

 in the manner in which tig. 2 already represents the second case 

 (p =. 30°. For the new problem , which makes its appearance in 

 the third case of the pentagon B 3 C 3 D 3 B 3 F 3 , viz. to find the 

 triangles (and hexagons) situated in the planes truncating the rhom- 

 bohedron, is easily solved. So in fig. 4, corresponding to that 

 case, the line WN' through W parallel to ON meets AC in a 

 point N' of FF and this construction always holds; in this special 

 case it is more to the point however to remember that E is the 

 midpoint of AD, etc. 



In fig. 4 the angle ANO indicates the inclination of the faces 

 of the section on the plane w of the hexagon // 6 . It is easy to find 

 the general relation between this angle ^ (fig- 1) and the angle 

 of rotation 0. From the two relations 



OR = O A cos Cp, OA= ON tang A> 



can be deduced 



tang vp = sec <p \/2. 



We lay down what we have hitherto found in the following: 



Table of results. 



FACES. 



Tans 



Ta no- 



An gl e 



Angle 



I 



II 



III 



IV 



V 



VI 

 VII 



6 squares A l B 1 C l D 1 (cube) 



6 lozenges A t B 2 C 2 D 2 



6 pentagons B 3 C t D 3 E 3 F 3 and 2 equilateral 

 triangles (side E 3 F 3 ) 



8 equilateral triangles #„£„/>„ (octah- dron) 



6 trapezia E S F Ù G & H 5 and two equiangular 

 hexagons (sides E s F s , G s f/ r> ) 



6 trapezia E s F e G e ff é and two equiangular 



hexagons (sides E G F„ , (<' c // r> ) 



6 rectangles (3 V 2 cm and 3 cm) and 2 It 

 (hexagonal prism) 







1 1/3 

 1/3 



2l/~3 



3 1 ~3 



l/~2 



fy/¥ 



2l/¥ 



1/26 

 2V/Ï4 



0° 

 30° 



40° 53' 36" 

 60° 



73° 53' 52" 



79° 6' 24" 



90° 



54° 44' 9" 

 58° 30' 54" 



61° 52' 28" 

 70° 31' 44" 



78° 54' 15" 



84° 42' 43" 



90° 



8. Before we can pass to the study of the block of 81 eight- 

 cells we must say a word about the section of a single eightcell 

 by a space normal to the plane t' but not passing through the 



