( 784 ) 



Vqi- 2 I,- — 1 a)id 2//] liave llic greatest comiriöii divisor (/. Am] for 

 odd /I the polygon is reduced in i(,n in this respect to a liiicsegnient 

 long- 2o hearing at one end the even numbers 0, 2, 4, . . . and at 

 the otlier the odd lumibers 1, 3, 5 . . /' 



"Let us replace for odd ii the just mentioned linesegment 2o by 

 a linesegment ^|/2 bearing at its ends the same groups of numbers." 



"Let us place for e\'en Jt the in planes and for odd n the ni — 1 

 planes and the linesegment (i[2 in such a way in the space >S,i that 

 in a common point they are rectangular to one another." 



"Then the // + i [or 2//] points /'; of that space the projections of 

 which on these /// elements coincide with the vertices numbered 

 with /. are the vertices of a regular polytope .1„ with length of 

 edges o\^ i/-\-r [ov ('„ with length of edges qY^n'].'' 



This (U»id_)le theorem where with respect to the coiiliuiious bifur- 

 catiou ■•fills [or l/nffT' we ninsi either always read this placed 

 before the lirackels or always th'it placed iiisitle the brackets, 

 reuiiiids ojie of the decomposition of the general motion in S„ into 

 /// c(Mn[)(»nents for e\eii // in /// rotations, foi' odd // in /// — I rotations 

 and a translation. This remark is important with resi)ect to the 

 decomposition of the grou[)s of anallagmatic motions belonging to 

 .1,. and ('„. 



T h e o r e m 11. 



"Let ,S' _i and S^, be two spaces rectangular to each other in a 

 point and let .S2/>— i represent the space determined by them." 



(1) 

 -Let us take in >S),_i a regular polytope J^,-i , m Sj, a regidar 



1) 

 polvtojte (],, having both as the index (1) indicates unity as length 



of edges.'" 



"Let us number the /i vertices of .l/,-i with the pairs of numbers 

 (0,y>), (I,y> + 1), {2,1, -\- 2),... ip-1, 2y^— i) and let us assign to 

 each of the 2j, vertices of (], one of the numbers 0, J, 2, . . . 2;j>— 1 

 under the condition that the j, tliagonals bear at the ends again the 

 pairs of numbers (0, p), (1, /> -f i), 2 (y> + 2), . . . {p—l, 2p~l)." 



"Then the 2p [)oints /^f of >%-^\, whose i)rojections on >S;,_i and aS/, 

 coincide willi the vertices of .l^.^i and C^ bearing the equal numbers,- 



form the \ertices of a i-euular polvtope Aop-i with length nt edges 



(1) 

 [/2 winch projects itself on .s^_i according to two coinciding A^-^i 



(1) ^^ 

 and on S^, accordijig to a ( ), ." 



By this simple theorem we are enabled to deduce the [)roof of 



