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Mathematics. — ''The section of the measure-polytope Mn of space 

 Spn loith a central space Spn—\ perpendicular to a diagorial." 

 By Prof. P. H. Schoute. 



(Communicated in the meeting of December 28, 1907). 



We determine the indicated section in three different ways : 



1. by means of the projection of 3fn on the diagonal, 



2. with the aid of the projection of Mn on a plane through two 

 opposite edges intersecting the diagonal, 



3. by regarding regular simplexes. 



I. The projection of Mn on a diagonal. 



1. We can easily prove both analytically and synthetically the 

 following theorem: 



"The vertices of the measure-polytope Mn project themselves on a 

 "diagonal in n -{-1 points, namely in the ends of the diagonal and 

 "in the n — 1 points, which divide the latter into n equal parts; in 

 "these n -\- i points are projected successively 



1, n, \n{n — 1), .... ^ n {n — 1), n, 1 

 "points, where these numbers are the coefficients of the terms 

 "of (aH-6)"". 



From this general theorem ensue the results for n ■=: 4, 5, 6, 7, 8 

 given in the diagrams added here (see the expanding plate). An 

 explanation of the sketch belonging to ?z ^ 4 will sufficiently explain 

 the others. 



The horizontal lines of this figure alwaj^s represent the same 

 diagonal on which the projection takes place ; on these ten lines are 

 successively indicated the projections of vertices, of edges, of faces 

 and of bounding bodies. In order to find space for the figures indicating 

 the numbers, the thick projection-lines have been broken off, where 

 such was necessary. 



Tf we designate the five points on the diagonal by a, h, c, d, e, — 

 S'^e the bottom line of the ten horizontal ones — then in these places — 

 see the topmost of the ten lines — 1, 4, 6, 4, J vertices are projected 

 there — bear in mind (1 + l)^ 



On the four equal segments ab, be, cd, de are projected successively 

 4, 12, 12, 4 edges — think of 4(1 -f l)^ 



In like manner the three equal segments ac, bd, ce are successively 

 the projections of 6, 12, 6 faces — think of 6 (1 -f- 1)". 



Finally on the two equal segments d, be are projected successively 

 4, 4 bounding bodies — think of 4 (1 -|- 1), 



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