DERIVED EROM THE REGULAR POLYTOPES. 7 



Here we have [1100]V"2 = C%„ [1000] V2 = 0%, [10000J V2 



U 32 . 



Remark. If we invert the sign of all the coordinates of a vertex V 

 of the polytope we get the coordinates of an other vertex V' of 

 that polytope for which the centre of the segment P P' is the 

 origin of coordinates 0. So, all the forms derived analytically from 

 the measure polytope admit central symmetry, as the geometrical 

 deduction by means of the operations e and c requires it. 



B. The characteristic numbers. 



50. In the case of the simplex the direct method for the deter- 

 mination of the characteristic numbers proceeding regularly from 

 vertices to edges, from edges to faces, etc. was preceded by an 

 easier method fulfilling the exigencies of the cases n = 4 and n = 5, 

 working from both sides, the vertex side and the side of the limi- 

 ting element of the highest number of dimensions; in this case of 

 the measure polytope we will do likewise. 1 ) 



Here also the number of vertices is easily found. If all the n 

 digits of the symbol of coordinates are different it is 2 n . n\; of the 

 two factors 2 H and n\ of this product the first is due to the 

 power of choosing arbitrarily the signs of the n digits, whilst 

 the second corresponds to the power of permutating them. This 

 product must be divided by 2! for any two, by 3! for any three 

 digits being equal, etc. 



In order to be able to find the number of the limiting bodies 

 (n = 4) and that of the limiting poly topes {n = 5) we have to 

 prove here the 



Theorem XXIX. "The non vanishing coefficients c t of the coor- 

 dinates tVj in the equation c ± œ v -j- c 2 #> -j~ . . . = p of a limiting 

 space S ll _ l of the polytope deduced from the measure polytope 

 of JS n must all of them have the same absolute value." 



, The difference between this theorem and the corresponding one 

 for the simplex (theorem II of art. 6) lies in the addition of the 

 word "absolute", therefore printed in italics. This amplification is 

 necessary here, in connection with the power of assigning to each 

 of the n digits of the coordinate symbol either the positive or the 

 negative sign. But the proof runs quite in the same lines. If in 

 the case of the polytope [1 -j- 2\/2, 1 -f- V2, 1 + V%, 1] we 

 start from the equation 2&\ — œ 2 =p and try to determine the 



l ) The treatment of the offspring of the measure polytope with which we are con- 

 cerned now — and of that of the cross polytope which comes next — will be copied 

 as much as possible from Section I, 



