DERIVED EROM THE REGULAR POLYTOPES. 97 



highest import of the third constituent is the contraction form of 

 the corresponding limit of the second constituent, i. e. the third 

 constituent itself is the contraction form of the second. Or shorter still: 

 by the reversion of the symbols the transition from (# 4 a 2 . . .a n _ 2 1—1) 

 to (ci A a 2 . . .a a _ 2 1 1) manifests itself by the diminution of the first 

 digit by 2, i.e. by the operation of contraction, leading to the 

 result mentioned in the theorem. 



Remark. There is a characteristic difference between the three 

 groups of nets — ■ a) the simplex nets, b) the measure polytope nets, 

 c) the half measure polytope nets — as to the character of the con- 

 stituents. As we have seen in the preceding sections the simplex 

 nets admit exclusively principal constituents, i. e. neither prisms 

 nor prismotopes, whilst the measure polytope nets admit only two 

 principal constituents with exception of the original net of measure 

 poly topes. Now in the case of the hmpd. net we always find three 

 principal constituents with exception of the original net (HM n , Cr n ); 

 as soon as two of the three constituents become equal to each other 

 we fall back on a measure polytope net. This only happens for 

 n > 3 in /# 4 , as we shall see in the next article. 



104. Hmpd. nets in 8 k . — Here we have to examine the eight cases: 



1 MM,, Cr k 



2 e 2 HM± , e ± Cr, 



3 e^RM,, e 2 Cr, 



4 e k HM^ e 3 Cr, 



5 e 2 'e z HM k , e i 'e 2 Q\ 



6 e 2 e k HM k , e x e 3 Cr, 



1 e 3 e k HM k , e 2 e- 3 Cr k 



8 ... e 2 e 3 e k HM k , e x e 2 e s 4 



Of these eight cases only four are new. The first is JV(C i6 ), the 

 three equal groups of (7 16 being the groups of -f- HM& — HM& Cr k . 

 The second case is e i N(C. i6 ); as e 2 HM [k = e i Cr k we find only two 

 principal constituents. The third case is ce 2 N[C x ^\2^e^HM k = ce 2 Cr k , 

 the third constituent is equal to the first. Finally the fifth case is 

 ce x e 2 N[C iG ) ; as e 2 e 3 HM k = ce x e 2 Cr ky here also the third constituent 

 is equal to the first. In the four remaining cases the three chief 

 constituents are different; so these cases are new. We represent 

 them in the following small table 



[ e^HM^ e 3 4 , c* 3 4 , P r "l, 2(1 + ^2) a t [2 + y%\ y 2 , V% , |/2], 



e e± HM± , e x e A Qr A , ce, e., Cr 4 , P tT ], 2(3 + y 2) a t [4 + y% , 2 + y 2 , y '2 , y%\ , 



e^e^HM^, e 2 e d Cr éi ce 2 e s Cr 4 , P T ], 2(3 + ^2)^ [4 + J/2, 2 + ^2, 2 + ^2, 1/2], 



\e % e. h e 4 KM^ e l e i e i Or ét ce 1 e z e B Cr é , P tT ], 2(5 + ^2)»/ [6 + J/2, *+y2, 2+y2, y2], 



enumerating the quadruplets of constituents and in condensed form 

 the net symbols; in latter symbols the immovable parts of the digits 

 are placed before the square brackets, whilst the sum of the four 

 integers a t is always even. 



Verh. Kon. Akad. v. Wetensch. Ie Sectie Dl. XI No. 5. E 7 



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