BY A SPACE ROTATING ABOUT A BLANK. 17 



rhombohedron or a part of it, obtained by truncation at one end 

 or at' both ends by planes normal to the axis; if we except the 

 special case <p = 90°, which will be treated separately, this rule 

 is general. Now this section can be designated by the symbol 

 a {p><?)> where a represents the length of the axis of the unmuti- 

 lated rhonibohedron expressed in cm. , whilst p and q indicate the 

 fractions of the axis cut off by the endplanes, the first of these 

 parameters refering to that endplane the external normal of which 

 corresponds in sense to A' A. For the unmutilated rhombohedron 

 is determined by the hexagonal section normal to the axis in the 

 centre, which section is the same for all the rhombohedra , and 

 the length of the axis. So a (0 , 0) stands for a whole rhombohe- 

 dron, a{p,Qi) or a (0 , p) for a rhombohedron truncated at one 

 end , a (p, p) for a rhombohedron truncated at both ends admitting of a 

 plane normal to the axis dividing it into two congruent halves; 

 so a(p, q) transforms itself into a(q,p) by a rotation through 

 180° about any line normal to the axis, etc. 



14. We consider the case III (tang cp = \\/o) in detail, and 

 to that end we reproduce in fig. 9 the projection of the block of 

 eightcells on ir' and the projection /= AB X C^. . .A' of the inter- 

 secting space. Here within the rectangle PP'Q'Q that is the pro- 

 jection of any eightcell of the group G 7 , let us say the central one 

 14, lies the segment C' 2 C 2 ' of /, whilst the points B^, B s ' on the 

 edges PQ, P'Q' are the endpoints of the axis of the rhombohe- 

 dron. So, as B> S C 2 = C 2 ' Z? 3 is a sixth part of B 3 B H ', Ave find for 

 the section \\/~tfî{\, J). 



In general the value of a is sec (pj/ö; in the following table, 

 which gives us for all the groups of intersected eightcells 1 st . the 

 segment of / lying within the projection, 2 nd . the segment of / re- 

 presenting the axis of the unmutilated rhombohedron , 3 rd . the sym- 

 bol a(p,q), this value of a is indicated at the head, which 

 allows us to use the simplified symbol (jp, q). 



Verhand Kon. Akad. v. Welensch. (Ie Sectie) D!. i\. n°. 7. ft 2 



