( 641 ) 



between the coefficients of the diftereiit powers of .^'in the development 

 of (1 -|- .V -f- ■»'" + • • • + ■''^~')^ a-J'd the inunl)ers of cells 6^-) of the 



block C^-^) which agree with each other in projection on a diagonal; 



In the foHowiiir»- table of results we have separated from one another 



i 1 



the three cases leading to sections - 6"^) = 7^21 -')- 6^2) — 6>C2^/2) 



4 ^ 2 ° 



1 3 



and the two cases leading to sections ^q2)_7\i2)^ -r;2 = .4'>^2; 



Moreover, the two positions of opposite orientation appearing for 

 T and A are distinguished from each other as 7),, Tn and A^,, A„, 

 and then those parts J'(i 2^ .^j,f| ^4(1^2) g^^ jj^g same foot-index which 

 answer not only as regards volume but also as regards position of 



juncture to the relation 



J (I 2) _|_ 4 7Xr-2) _ 7X3K2) ^ 



whilst this i]idex is a p (positive) for T<-^-^ when this tetrahedron 

 agrees in position to 7'(2/.i'2) ^y^^\ ^4(2Ai 2)^ ^,^(i ^.^^j, 5^ taken arbitra- 

 rily in the third case -^'-^-\ where the two amounts are indeed 

 equal. 



In this table the symbols {k -\- 2)^, etc. represent binomial coeffi- 

 cients. The coming to the fore of the numerical factor 23 is 

 connected with the relation holding only for the volume 



_^4(K2) — 23 7"i^'-\ 

 which ensues immediately from the one given above. It forms part of 



(9(2K2, ^(1/2) 7'C2K2) 7^(»/2) 



32 ~" 23 ~ 8 ^ J ' 



of which we have availed ourselves when arranging the preceding 

 table, either as an aid in the calculation or as control. 



The cases (1, 1, 1, 0), (1, 1, 0, 0), (1, 0, 0, 0). — These three cases are 

 so much simpler than the preceding one, that we can treat them 

 collectively, now that the application of the results appearing here to 

 the nets {C\^) and (C^J make a short treatment necessary. The pro- 

 jection of the bounding elements on the corresponding axes OK, OF, 

 OR are immediately found ; in order to take into account the duality, 

 appearing on one hand between OE and OR and on the other hand 

 between OK and OF, the projections on OR are placed on the 

 first plate next to those on OE, whilst the projections on OK and 

 OF find a place there side by side. A single glance given to these 

 diagrams already arouses the conviction that the sections in the direc- 

 tion of BE over OK and OF to OR must keep on becoming sim. 

 pier. That this is really the case — and for what reason — is 



44 



Proceedings Royal Acad. Amsterdam. Vol. X, 



