NM& = 



C^a 1 6fji 6 C - . 



. . e, e, e, e n NMf\ 



NM.™ = 



e a' e b' e c' 



• ' ' @r' *s' ^l' "n LV -M- n , 



NM™ = 



Ce a' C b' e c' • 



..e r .e,e, NM™, 



3(5 ANALYTICAL TREATMENT OF THE P0LYT0PES REGULARLY 



EN = cE'e n N' , Ee n N = E'e n N' , cEN = cE'N' , 



where N and N' represent polarly related regular nets of JS n , whilst 

 the sets of operations e ki [k = 1, 2, . . . , n — 1), contained in E and 

 E' are complementary to each other, i.e. that E' contains the ope- 

 rations e n _ k complementary to the operations e, c of E and no other 

 one. Now, in the case of the net of measure polytopes we have 

 N' = N; so we get: 



Theorem XLT. "We have the relations: 



^a^b^c • ' ' &r ^ s ^t 

 ^a^b^c ' • • ^r ^s ^t ^n 

 c @a ^b^c ' ' • &r ^s e t 



under the conditions 



a -\- 1' = b -f- s' = c 4~ r = . . . = r -\- c' = s -f- b' = t -j- a == n ; 



then the constituents g Q , g 1} g 2 , . . . , y rt _ 2 , g n -±i [In °f the one are equal 

 to the constituents g' n , g' n _, v , g' n _ 2 >> • -, 9^ /i, ƒ o of tne other. 

 So the nets e a e h e c . . .e r e s e t e n NM n (2) and ce a e b e c . . .e r e s e c NM n (2) 

 are semiperiodic under the conditions 



a-\-É=ô-\-s = c-\-r = ... = ?z. 



In the latter cases there is an unpaired middle constituent for n even." 

 Proof. We prove each of the three relations by showing that the 

 extreme constituents g ,g n °f the ne ^ a ^ the ^ e ^ °f the sign of equality 

 are equal to the constituents g' n ,g'o of the net at the right. But 

 we suppose that it will do to enter into details for one of the 

 three relations, say the second. 



In the case of the net e a e b e c . . . e r e s e t e n NM n (2) , where as in 

 art. 38 we suppose the indices of the h -\- 1 factors e a ,e b ,. . . e n 

 to be arranged according to increasing values of the subscripts, 

 the principal constituent g n is, according to theorem XXXV: 



n—t t—s s—r 



[k',k',..k', {k—\)',{k—\)',..{k—\)', (k— 2)',(k— 2)',..(k— 2)', . . . , 



c—b b — a a 



â 7 , 2', . . 2', r,r,..i', i,i,. .1]. 



So we find according to theorem XXXIX for g by subtraction 

 from k' -\- 1 : 



a b— a c — 



[k', k\ . . k', Jf-\)',{k—\)',..{k—\)', (i— 2)',(&— 2)',..{k— 2)', .... 



s—r t — s n—t 



2',2',..2', l',l',..l', L.1....1]. 



