( 390 ) 



four edges of a Cs meeting in a vertex P, if PQ and PR are the 

 squares described on XJ J X 2 and X t PX t and if F, G, are the 

 centres of these faces and of the eighteen, FPGO is a square and 



the net (G' s ), to which the cell C%~ belongs, projects itself on the 

 plane of the square PR as a plane-tilling of squares (lig. 7), whilst 

 the intersecting space normal to the diagonal ]*R of the square pro- 

 jects itself in a normal to that line. We expressed this in the second 

 communication by saying that the problem of the section of a four- 

 dimensional polytope by a threedimensional space has lost here two 

 of its dimensions. If now amongst the lines normal to the diagonal 

 PR line a passes through R, line h passes through the points S lt 



1\ on the sides RS, RT for which RS 1 ~ — RS, RT l = — RT, 



4 4 



and line c passes through the midpoints S„ 1\ of RS, RT, then the 

 position a of the intersecting space corresponds to the first and the 

 fifth figure of the fourth column of plate II, the position b corresponds 

 to the second and the fourth figure, the position c corresponds to the 

 third one. 



A second remark refers to the position of the sections obtained in 

 the third and the fourth column. The first and the third figure of 

 the third column are equal to the first and the fifth figure of the 

 fourth column; also the middle figures of' the two columns are equal. 

 But there is a difference in position. In the figures of the third 

 column the axis MN of period four is vertical, in the figures of the 

 fourth column this axis MN is horizontal. This is not accidental. 

 As both columns represent the sections with the polarly circumscribed 



Cg' and the bodily circumscribed Co in the same orientation, it 

 proves that the axes MN of the sections of the erect sixteencells 

 and those of any of the two groups of the inclined sixteencells are 

 normal to one another, from which may be derived that the three 

 axes MN of the sections of three sixteencells, any two of which 

 belong to different groups, are normal to one another by twos. We 

 verify this by proving that the axes MN of the sections of two 

 inclined sixteencells of different kinds are normal to one another. 

 Therefore we remark that the limiting spaces P(X l X t X A ) and 

 P(X,X,\\) of fig. 6 are parallel to OF - as the line GP parallel 

 to OF lies in P(X 3 X 4 ) - - and so the intersecting spaces normal to 



OF are normal to those limiting spaces of C\ . As the sixteencells 



inscribed in C$ and in an adjacent Cs J are one another's mirror- 



