REGULAR FOUR-DIMENSIONAL HYPEESOLIDS. ] 1 



/(./ ) and S intersect in a plane passing through the points D' 

 A {A B), A \a). 



a (I I, 



A projection of this on a plane through A I)' {A S) A is given 

 in tig. 19. There D'r is the projection of the plane of inter- 

 section of /./ and S ; A ?' k made equal to ^; « and the line i)V 

 divides {A B) A in the ratio a-. I — r/; /^r is drawn through (J 5) 

 and is parallel to I)' r. Then we have r (A J =^pa. 



Also let the length of which D' T' is the projection be equal to c 



times the edge, from which we see that /S" passes through the points 



(A ) A, (A ) A, D' iD' ),D' iD' ). In diagram II these points are 



2, 1, 14, 11 ; () and 5 are {A B) and {A C). A plane through I A 

 parallel to i? 6' j9 and passing through the points (A B), (./ C), (AI)) 

 gives the points i>' {A ), B' (A ), C' {A ), the first of these being 



point 8 of diagram II, fig. 1 1 , (see list of vertices). 



This model is constructed in the same way as the first. The 

 points where the sectional space cuts the edges of the 000-cell are 

 determined by means of the icosahedra ^) and are then found on 

 the individual teti'ahedra. 



12. Let the GOO-cell be cut by a, space /S' through(.7 ),(./ J, (^/ ), 



(7i p, {B^X {B^), (6'p, (6y, (C;;, {D^), {IJ^X {D^y, this is parallel to s] 

 and its intersection with / (J ) is given in projection by the line 

 V {A ) parallel to Vr ^fig. 19). Nuw V bisects D\D' ), and let 

 {A B) ShQ e([ual to ƒ times the edge. Again V {A ) divides (A. B){D' ) 



in the ratio a: 1 — a. We have then the points {A B) , {A" B), 



{AC) {/ C),{AB) (JJ'),{A C) {D' ), I)'\ (Z>' ), B' \ {JJ' )■ see on 

 diagram III the points (i, 5, 13, 14, 18,8; 1 and 2 are (.-^ ) and 

 {A ) (consult list of vertices). 



13. Let the OOO-cell be cut by a space /V^* through (J' ), (^4'^.), 

 {A^, (7/^), {B\_X{B\), (6"^),(C'^),(C"^),(i)';,(i'p,(i);,);thisisparal- 



lei to S . The intersection of /S\ and I {A ) is shewn in projection 

 by the line (X>' .) >-^ parallel to V {A), fig, 19. 



') See fcot note on page 8. 



