DERIVED FROM THE REGULAR POLYTOPES. 101 



6+ 1/2,4 + ^2, 2 + 1/2,1/2 

 10 -f 2V2 ,0 , , . 



subtr. 



subtr. 



— (4+ V"2) , 4 -fV/2 ', 2 + V2 , V2 

 furnishing the poly tope [4 -[- V/2, 4 -f V" 2, 2 + V 2, V/2], i.e. C; 



6 + V/2 , 4 + V/2 , 2 + V/2 , N/2 



5 -f V/2 , 5 + V/2 , 5 -f V/2 , 5 -f V/2 



leading to the polytope — ^ [531 1] , i.e. A; 



6 + V/2 , 4+V2 , 2-+-V/2 , [V/2] 

 5 4-V2 , 5 + V/2 , 5 4- V/2 , "O 



-I ,-1,-3 , [V/2] SUbtr ' 



giving finally the polytope -^ [3 11] [V/2] , i.e. D. 



This will be clear, if we only add one word about the factor 

 J before the symbols of A and D, viz. that we want this 

 factor in order to have symbols representing polytopes with one kind 

 of vertex and one length of edge. 



106. Hmpd. nets in S h , — We have determined the sixteen limpd. 

 nets of 8 5 by means of the two methods given in outline in the 

 preceding article. 



The results of the first method are put on record in Table X. 

 This table is divided by vertical lines into eight parts; of these 

 the first contains the symbol of the nets, the last two their consti- 

 tuents and the five others the limits (7) 4 of each of the five 

 constituents A, B , C, D, JE. In the construction of this table we 

 started from theorem LXVII enabling us to register in the last 

 part but one in the columns with the superscripts A,B, C the 

 character of the three principal constituents and to add under J9, 

 in the cases where e k appears amongst the expansion symbols of the 

 net, the prisms on the polytopes of polytope import of A as bases. 

 After having finished this task we have inscribed in the columns 

 with the headings A$,A t ,. . .~D t , B the limits (/) 4 of these consti- 

 tuents A, B , C, D, taken from the tables given in the preceding 

 sections of this memoir; this will be clear if Ave add the remark 

 that the notation I) 3 , D t , D for the limits (/) 4 of D differing from 

 the bases _D 4 has been chosen in accordance with the consideration of 

 these bases as deduced from HM k . This second task having been 

 performed we can formulate the contact between the constituents 

 A,B,C,D; we find generally: 



A k = A k (if <? 4 is absent) and A k = D k (if e k is present), 

 A t = B k , A = 6 4 , x>3 = D 3 , By = x>! , B = 6 , 6 3 = i) . 



