92 ANALYTICAL TREATMENT OE THE POLYTOPES REGULARLY 



For we can start either from HM n as it is, or from one of the 

 (n—\\ forms e ki HM ni or from one of the (n — 1) 2 forms e k e k HM )n 

 etc., giving altogether 



1 + («— 1), + («— 1) 2 + + (•— 1) M _! = (1 + If" 1 = 2- 1 



possibilities. These nets must all be new for n > 4, if they prove 

 to exist. On the other hand a preparatory study of the cases n = 3 

 and n = 4 will show that # = 3 furnishes nothing new, whilst 

 n = 4 produces four new cases only. 



102. Hwpd. nets in S s . — If we interprète the net of T and O 

 as iV(+ HM à , Cr 3 ) the four cases we meet here are 



1 . . . . BM„ Or, 



2 . . . . e 2 HM%, e v Cr d 



or in other form 



1 T, O 



2 tl\ W, . . CO 



12 



24 



3 . . . . <? 3 HM iy e 2 Cr % 



4 .... e 2 e 3 RM 3 , e^ e 2 Cr- S 



3 T,HCO, . . C 19 



4 tl\ WO, . . W 23 



Here the third constituents CO, C, tC are polyhedra filling gaps, 

 whilst the numbers 12, 24, 19, 23 refer to the stereoscopic 

 diagrams of Andreini. Compare also Table III of M rs . Stott's 

 memoir. 



Let us pass now to the deduction of the coordinate symbols of 

 these four nets. To that end we have to start in the first case 

 from a T and an O in face contact — and in the other cases 

 from what these polyhedra have become — and to calculate by 

 means of the distance of their centres the periodic term which is 

 to figure in the symbol. We therefore elucidate the mutual position 

 of the two polyhedra in face contact in fig. 19, in projection on 

 to a plane normal to one of the three diameters of the O group. 

 But for clearness' sake we have represented in each of the four 

 cases the T and the O — or what they have become — lying 

 apart; in order to re-establish the real state we have to move the 

 T parallel to itself so as to bring the invisible shadowed face of 

 T indicated by dotted lines in contact with the visible shadowed 

 face of 0, i. e. A' B' into coincidence with AB. As we want 

 only the net symbol with respect to the group of 0, the origin 

 of the system O(XYZ) of coordinates has been chosen in the centre 

 of the O of the diagram. 



The simple diagrams of fig. 19 show an easier way leading to 

 the knowledge of the periodic term of the net symbol. Indeed, in 

 each of the four cases the O — or what it has become — is in 



