70 ANALYTICAL TREATMENT OP THE POLYTOPES BEGULAltLY 



Theorem LTV. "Any poly tope (P) n of cross poly tope descent in 

 S n has the property that the vertices V i adjacent to any arbitrary 

 vertex V lie in the same space S n _ x normal to the line joining this 

 vertex V to the centre O of the polytope. The system of the spaces 

 8 n _ i corresponding in this way to the different vertices V of ÇP) n 

 include an other polytope (P)' n , the reciprocal polar of (2 J ) n with 

 respect to a certain concentric spherical space. But in the case of 

 the cross polytope itself these spaces pass through the centre.'' 



This theorem is a mere transcription of theorem XL. 



87. If we apply the general relations of polarity, which have led 

 us in art. 67 to theorem XLI, to the particular case of the polarly 

 related nets J\\C iG ) and J\\C, k ) of # 4 we get: 



Theorem LV. "If the sets of operations E and E' are comple- 

 mentary to each other, i. e. if E' contains the operations e^_ k comple- 

 mentary to the operations e k of E and no other, we have 



EN{ C i6 ) = cE'e, N{ C 24 ), Ee, N{ <7 16 ) = E'e, 7V(£ 24 ) , cEN{ C i6 ) = cE'N{ <7 24 ), 



cEe,N(C i6 ) = E'N(Cu)" 



An analytical proof of this theorem would require a more ample 

 knowledge of the net symbols of the nets deduced from N(C 2i ) than 

 we have at our disposal, after having nearly finished the third part of 

 this memoir. We will therefore invert the order of ideas, i. e. we will 

 content ourselves here by giving the analytical form of the geometric 

 facts and use theorem LV and the last column but one of Table VII 

 based on it in the last section of this memoir dealing with the extra 

 regular polytopes, to facilitate and control the deduction of the 

 poly topes and nets, deduced from G. lk . There we shall have occasion to 

 apply the same principle to the polarly related polytopes C m and C 120 . l ) 



88. The connection between C s , C i6 , C 2k according to which the 

 (7 24 (2) can be split up with respect to its vertices iuto a C s (2) and 

 a 6 r 1(i ( 2 ^ ;2 ) and with repect to its limiting spaces into a C 8 (2j/2) and 

 a C v - n leads to connections between the polytopes and the nets 

 which cannot be deduced from polarity only. So we find: 



(7 24 = ce 2 C 8 (= ce x (7 16 ), e i C 2k = e i e 2 C iQ 

 and 



N(C 2i ) = ce, N(C m ) = ce, N(C i6 ) = ce d N(C n ). 

 But there is still a more striking coincidence to be indicated, viz. 

 that the nets e 2 N(C i6 ) and e x e, N(C ie ) are respectively equal to the nets 



l ) We defer the investigation of the reciprocal nets of those given in Table VII to 

 the paper announced in the foot note of art. 68. 



