40 ANALYTICAL TREATMENT OF THE POLYTOPES REGULARLY 



this new version, unless we prove that each net of the assigned 

 kind does admit a net symbol ; as soon as such a net admits a 

 net symbol we can choose our system of coordinates in such a 

 manner that this net symbol contains integer coordinates only. We 

 take position with respect to this point by supposing beforehand 

 that each net of the assigned kind admits a net symbol, which 

 brings us under the obligation to prove afterwards that this is so. 

 b). Are there simplex nets not satisfying the condition that all 

 the constituents are of simplex extraction ? 



We dispose of this question by pointing to three plane nets, viz. 

 1°. JV(p s ,pr l , p^ of triangles, squares and hexagons (fig. 6), 

 2°. JVip'^Po' P12) 0I ' squares, hexagons and dodecagons (fig. 7), 

 3°. N(p 6 ,p vl ) °f triangles and dodecagons (fig. S), 

 which must undeniably be considered as simplex nets , as they can 

 be derived from the three generally known plane nets JV(p- s ), N(p 6 ), 

 N(Ps,Pe) by means of the ^-operations. If N(p x ;p y ',p^) represents 

 a net with the polygonic constituents p x ,p y ,p z of which p x is of 

 face, p of edge, p z of vertex import, these deductions are indicated 

 by the equations 



e-i N{p£) = N (p. à ; p k ; p 6 ) ; e 2 N (p 6 ) = N (p 6 ; p k ; pj ; 



e { e 2 N (;/ 3 ) = N (p 6 ; /? 4 ; P12) ; ^1 e 2 N (p 6 ) = N ( p n - p k • p 6 ).; 

 c e i e 2 iV (p s ) = N(py, — ; p i2 ). 



As these three nets contain constituents not deducable from the 

 simplex of the plane, the triangle, by means of the operations e lc 

 and c, they must form exception to the general rule about the net 

 symbol with integer coordinates only ; for, in the coordinates with 

 respect to the simplex, only the polytopes derived from the simplex 

 can be represented by a symbol with integer coordinates only. 



On account of the property of the three plane nets mentioned — 

 to admit at the same time constituents derivable and constituents not 

 derivable from the simplex — we call them "hybridous". In order 

 to be able to deduce general results from the simple law of integers 

 found above we discard provisionally the three hybribous plane nets 

 and all the hybridous nets that space and hyperspace may contain, 

 considering only the nets we call simplex nets "proper"; meanwhile 

 we promise to come back to these exceptional cases, after having 

 secured the general rule alluded to in art. 22 and the main results 

 to which it leads (see art. 34). 



25. In the second place we remark that the two sets of triangles 

 of the net JV(p 3 ) admit the same frame symbol with integer coordinate 



