G G ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



[ (^(1 + 1/8)4-1, 0,(1+1/8) +0, fli(l+l/S)+0)^&i« = ? 

 ^(ft;A;ft)-. (^(l+v/3) + iv/3 + 0, ^ 2 (l+v/3) + |v/3 + l, 



l «3 (1 + 1/8) + -J 1/8 + 1), So, = — 1 . 



NfaiPtiPÙ... («i(l+iV8) + 8, « 2 (l+il/8) + l, 



flb(l+il/8) + 0) f Zfl l = 0. 

 X(p 6 ;p,',Pi2).. («i(l+v'8)+2 f ^(l+V8)+i,flii(ï+|/8)+0) f 2ft=0 > 



the 0j different from each other with respect to mod. 3. 



((« 1 (2 + v / 3) + l J r/ 2 (2 + v / 3) + 0^3(2 + V/3) + 0),Z^ = 0, 

 ^(ft;— ;*«).. (K+iV3)(2+y3) + 0, (% + -JV8)(2 + i/3) + 1, 



' (fl 3 + i 1/3) (2^+ 1/3) + 1), S*, = — 2. 



In space we find two hybridous nets. If N(A; B) represents a 

 net with the polyhedric constituents A,B, the first being of body, 

 the second of vertex import, these two nets and their generation are 

 indicated by the equations 



e 2 JV(T, 0) = N{T, BCD; C), e x e 2 N{T, 0) = N(tT, WO; tC), 



the stroke under referring to this that the expansions are to be 

 applied to 0. Here we even abstain from mentioning decomposing 

 symbols. 



Which prospect opens hyperspace for the hunting up of hybridous 

 simplex nets? Very probably none at all. For the most powerful instru- 

 ment in the plane, the operation e n , is quite ineffective in ordinary 

 space already, whilst the two hybridous nets of that space are due 

 to the special character of the octahedron as simplex polyhedron. 



Nets with two kind of vertices. Neither is it probable that hyperspace 

 contains nets with a constituent occurring in such a manner in two 

 different orientations that any vertex of the net only belongs to one 

 poly tope of one of the two sets; for in JSf 2 and in JS 3 the only nets 

 admitting this particularity are precisely hybridous nets, the net 

 N(P-3 >Pi2) with respect to p z> the net e 2 N(2\ 0) with respect to T 

 and the net e 1 e 2 N(T 3 0) with respect to tT. 



35. We finish this paragraph by mentioning other generations of 

 the nets n f and n n of the theorems XVIII and XIX. 



"If we start from a simplex S(n + 1) (1) of JSf n and complete the 

 n + 1 spaces /S / n _ 1 bearing the n — ^dimensional limits tf(») (1) to 

 n + I systems of equidistant parallel spaces # n _ ls the distance between 



