32 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



that by interchanging the two extreme forms with one another the 

 intermediate constituents return in inverted order of succession. 

 This remark suggests an answer to the question raised just now. 

 By taking the constituents g n ,g n — \,. . . , g\ , </ contained in the 

 second, third, ...,«-{- I st , n -f- 2 nd column of the same horizontal 

 line corresponding to a certain net in reversed order of succession 

 we get the constituents g' n ,g n — \,. • -,g\,g\ of a net bearing in 

 general an other name, the operators occurring in which are inscri- 

 bed in the n -f- 3 rd column ; this net with constituents with com- 

 plementary import is essentially the same as the original one. So by 

 inverting the order of succession of the imports the three groups 

 (e, c), (e, e), (c, c) pass into (c, e), (e, e), (<?, c), in other words the 

 first group furnishes the group (c, e), whilst each of the other groups 

 passes into itself. We have used this fact, to which we shall have 

 to come back in part F of this section, in order to simplify the 

 Tables V and VI. So on one hand the nets (<?, e) have been omitted 

 totally, whilst on the other the number of lines of the groups (e, e) 

 and (c, c) have been diminished by writing down the nets in a trans- 

 parent systematical order and omitting at any time the net appearing 

 already in inverted order under the preceding ones. 1 ) 



In the column under the heading p. some particularities of the 

 nets have been inscribed. By r. we have indicated that the net is 

 regular, by s. p. that it is "semiperiodic", i. e. that the two extreme 

 forms are the same which implies the equality of any two consti- 

 tuents with complementary import. 



The other columns will be explained later on. 



A survey of the results contained in the tables suggests the 

 following remarks : 



a). There is a great difference in character between the consti- 

 tuents of a simplex net proper on one hand and those of a measure 

 poly tope net. All the constituents of a simplex net proper are expan- 

 sion and contraction forms of the simplex, whilst we found just now 

 that in a measure polytope net in general only two of the consti- 

 tuents, the groundform and the opposite form, are expansion and 

 extraction forms of the measure polytope. 2 ) 



*) The cases ce 2 N(C 8 ), ce 3 A 7 (C 8 ), etc. do not figure in the first third part of Table II 

 contained in the memoir of M rs . Stott, as they appear already as expansion forms 

 under either 2V(C 10 ) or N(C 2ll ). 



In order to spare room we have omitted in Table VI the column containing the name of 

 the net taken in inversed order. For the upper and middle part it is always the symbol 

 before M. under f/ Q to which e 5 has been added, for the last part it is that symbol itself. 



2 ) Compare for the prisms and prismotopes entering here my paper: "On the cha- 

 racteristic numbers of the prismotope", Proceedings of Amsterdam, vol. XIV, p. 424. 



