DERIVED FROM THE REGULAR POLYTOPES. 37 



Likewise we get for the constituents g' n and g' of the second 

 net represented by e a . e b , e & . .e r . e s . e v e a N M n {2) the same expressions 

 in which the a, b, c, . . .r, s, t are dashed. From this it ensues that 

 we shall have at the same time g' n =g' Q and g' =g n under the 

 conditions 



a = n — t' , b — a '= t — s' , c — b = s — r , . . . 

 s — r = c — b' , t — s — b' — a , n — t = a' , 



giving immediately 



a + t' = b + s = c + r = . . . = r -f- c — s -f- b' = t -\- a = n. 



These conditions pass into 



'a-\-t=h-\-s = c-\-r=. . . = n , 



if the two nets coincide in a semiperiodic one l ). 



Remark. If we count as one the two nets which pass into each other 

 by interchanging the two extreme forms (and also the two nets iVand 

 e n N of measure polytopes only) the number of measure polytope 

 nets is 8 + 2.5 = 18 "in # 4 , 16 + 2.9 = 34 in # 5 , 32 + 2.19 = 70 

 in S 6 , 64 + 2.35 = 134 in S lf 128 + 2.71 = 270 in S s , etc. 



68. The circumstances under which polarization of a measure 

 polytope net leads to an other measure polytope net are easily 

 indicated. For, though in the case of a net belonging to the family 

 (e, e) the centres of all the constituents are the groups of centres of 

 the different limits (/) , (/),, (/) 2j ..... (/) n _ ls (/),, of the net N(M n 2m ), 

 m being the extension number, and these points form together the 

 vertices of a net N(M n m ), it is only N(M t f) itself which satisfies 

 the condition that an M n {2) the vertices of which are centres of the 

 M} 2) of the net includes only one vertex of this net. So, if we 

 discard the case ce 2 iV(J/ 4 ) = iV(67 24 ), the net N{M n ) and the one 

 deduced from it by polarization form together the only pair of tioo 

 reciprocal nets of measure polytope descent. 



In general the system of vertices of a net obtained by polarizing 

 a measure polytope net is the combination of several groups of 

 centres of limits M'£ 1,n) of the measure polytopes of the net N(M 2,n ), 

 m beino* the extension number. So we find in 8*: 



J ) la the case of the first relation, where we do not obtain the second member by 

 dashing the subscripts a, &,c, . . ., r, s, t of the first, the proof is a bit more complicated. 

 Here we find for g n the expression given above, but for g — as we have to subtract 

 from /c x instead of h\ -f- 1 — 



a b — a c — b s — r t — s n — t 



\ k,Ji,.. V, k— l,fc— 1, ,.k— 1, k-2,k— 2, . . A— 2, . ..,2/2,.. 2, l,l,..l^ 0, 0, .70 ] J/2, 

 etc. 



