( 542 ) 



and when one compares the rectangiüar fivecell of C^^^'^) to tlie 

 Cip of C^^^ —, and the erect C,, together fill a third of Sp,. 



In the second mode of transformation of the cells C^^'> of the net 

 (Cs) into the cells C^^ 2) of ^ ^et (C,,) the vertices of the C^^^^) 

 concentric to C^^) are the centres of the faces of these C^?^ , from 



8 



which it follows that the six centres of tlie faces of each of the eight 

 bounding cubes of C^^' are vertices of a bounding octahedron of 

 C^^-^ and so this cell may again be called inscribed — and bodily 

 inscribed too — in C^). Also the remaining bounding octahedra can 

 be directly indicated with respect to these circumscribed 6'^-); through 

 each of the sixteen vertices of O^' pass six faces of this cell, of 

 which the centres form the vertices of a bounding octahedron 



of c<y^\ 



From the preceding it follows, that the fourdimensional cases, in- 

 closing the cells C^^ ^^ and having edges parallel to the axes of 

 coordinates, consist of two nets (Cg) of cells C^-), which by exchange 

 of centres and vertices pass into each other. 



6. We conclude this first part by indicating the connection 

 existing between the systems of axes of the five dilFerent cells 

 with the origin of coordinates as common centre, which can be 

 obtained by parallel translation of one of the cells C-^), one of each 

 of the three groups of cells C ^f-^ and one of the cells C^^^2\ We 

 indicate these cells for brevity by Cg, C'lg, C\^, C'^, C^^ where 

 C16 represents the polarly inscribed sixteencell and C\^ and C"i, 

 successively the positive and the negative bodily inscribed one. Further 

 here too — according to the notation of the handbook mentioned 

 above — E, K, F, R will denote a vertex, midpoint of edge, centre 

 of face, centre of bounding body and therefore OE, OK, OF, OR 

 will have to denote the axes converging in these points. Thus OE^ 

 is an axis OE of Cg, OK^^ an axis OK of Cjg, 0F\^ an axis OF 

 of C'le, etc. 



The numbei's of axes OE, OK, OF, OR of each of the three 

 different cells are always the halves of the numbers of the elements 

 E,K,F,R; they are contained in the following table. 



Here C,e of course represents the three cells Cj,, C^, C'l,. 



We now indicate the connection of the svstems of axes of the 



1) By doubling tlie radii vectores of the six centres uf the faces from the chosen 

 vertex of these Cg '' we find the central section normal to the diagonal of this 

 point. 



