DERIVED FROM THE REGULAR POLYTOPES. 95 



to the rule connected with the sum of the digits would be 

 1, 3, 1 -f f V2, 8.+ f V2 in the four cases. 



103. The four cases of hnpd. nets in 8^ considered above agree 

 in this that the third constituent is the contraction form of the 

 constituent of octahedron origin. Indeed the contraction forms of 

 0, tO, RCO, tCO are respectively a vertex, CO, C, tC. This fact 

 is too general to be accidental, we will show why it must be so. 



Therefore we recur to theorem LXVI. As all the vertices of the 

 net figure in the net symbol of the octahedron group — which 

 implies as we already remarked that all the vertices of the new 

 constituent are contained in the net symbol — , the faces which 

 that new constituent has in common with the adjacent polyhedra 

 of the octahedron group must define that new polyhedron. Now in 

 the original net {O, T) any vertex V is a point of concurrence of 

 six 0, the centres of which are the opposite vertices V / i of the 

 six edges of the net of cubes from which (O, T) has been deduced. 

 So the six faces of contact of the new constituent with the six 

 polyhedra of octahedron origin lie in planes normal to the lines 

 OV i9 in the centres of these faces, lying at equal distance from 

 0. These simple considerations lead to three possibilities compatible 

 with the condition that the new constituent must admit vertices 

 of the same kind and edges of the same length: either the new 

 constituent is equal to the constituent of octahedron origin, or the 

 new one is the contraction form of the other, or the other is the 

 contraction form of the new one. But the first and the last sup- 

 positions are to be rejected. For the first would bring equality 

 between the two kinds of limits of the constituent of tetrahedron 

 origin which have been called original limits and limits of trun- 

 cation import, whilst the last is inadmissible as the constituent of 

 octahedron origin is no contraction form. 



We now prove that the preceding result holds for any hmpd. net 

 in jS h . If once for all we distinguish for short the constituent 

 of HM n origin as the first and that of Cr n origin as the second 

 we can extend theorem LXV by proving : 



Theorem LXVII. "Any hmpd. net has three different con- 

 stituents, none of which is a prism. The third is the contrac- 

 tion form of the second. So, if the first is e k e,. . . .e k e,. HM„ 



and therefore the second e,. 1 e,. a. . .e k a e k _a Cr n , the third 



is ce,, a e h *. . .e h x e,. A Cr„. In this form of the statement 



each of the three unequivocally determines the two others." 



