( 486 ) 



It is ensy to deduce from this tlie results given in the otliei 

 diagrams for ?z = 5, 6, 7,8, if we keep in mind, that the coefficients 

 by which (1 + l)^ (J + i)^ (1 + l)^ (1 + 1) are multiplied are 

 1, 4, 6, 4 and so by addition of unity at the end pass into a repetition 



of (1 -f l)^ 



2. More generally holds the following theorem, comprising the 

 preceding: 



''The vertices of each bounding Mp of l/„ {p ^ n) are projected on 

 "the diagonal of J/„ in j)-]-! successive points of division of that 

 "diagonal; here again the projections are distributed according to 

 "the coefficients 1, p, k P ip — !),••• of {a -f- b)'' over these p -\- '1 

 "successive points." 



The vertices of a bounding square are projected in three of the n-\-l 

 points, which naturally demands the division 1, 2, 1. The vertices of a 

 bounding cube are projected in four of the ii -\- 1 points, which of 

 necessity must lead to the division 1, 3, 3, 1 as by the preceding 

 the division 2, 2, 2, 2 is excepted. 



From this ensues then directly the following theorem: 



"The section of a space Spn~\. perpendicular to the diagonal of Ji» 

 "forming the axis of projection, with the space >S);^ bearing a bounding 

 "Mp of Mn is an Spp—i in Spp perpendicular to the diagonal of 

 "3fp connecting the two vertices of Mp projecting themselves in the 

 "ends of the projection of Mp." ^) 



But there is more. If p' (Mj,) represents the section of a measure- 



polytope J\fp with a space Spp—\ of its space S2)p perpendicular to 



one of its diagonals in a point of which the distance to the centre 



1 

 of the diagonal in the diagonal as unity amounts to — — j^'^ from 



1 



which is evident that p' ^-, the two theorems hold : 



"For even n a bounding measure-polytope 3Ip of J/„ is intersected 

 "by the central space 82^,1—1 perpendicular to the diagonal of M„ 



1) The indicated diagonal dp of Mp is the projection of the axis of projection 

 d on the space SjJp of Mp ; so we can obtain the projections of the vertices of 

 Mp on d by projecting these vertices first in Sj^p on dp and projecting afterwards 

 on d the points found on dp by the preceding means. 



As dp and d in the edge of Mn as unity are represented by Kp and ^^n and 



P 

 dp is projected on d as - of d, the cosine of the angle between d and Spp is 



equal to - i/wp. 

 n 



