( 785 ) 



theorem I fui' the polytope C,, out uf the one for the polytope An. 

 Bj repeated application avc arrive at: 



, 9 



"A .1,, of the space S ,, , can project itself on a definite 



2' 1 -■' — I 



system of multially rectangnlar spaces /S'.,y-i , S^j^ ■>,... S , S^, S 

 respectively according lo a ^'„_i , two coincided 6',._2 ,•■. 



(1/2) (1) 



2'/— 3 coincided C4 , 2'/~- coincided squares 62 and 2'/— 1 coincided 



(1) 

 linesegments 6'i ". 



Mathematics. — "(>^m sipniiietric transforinatloii of S^ In connection 

 Willi Sr tfud Si" By Mr. L. E. J. Brouwek. (Communicated 

 by Prof. D. J. Korteweg). 



Let us foi' I lie present occupy ourselves with a particular case of 

 symmetric transformation - — tiie rcfiectton, and let us investigate its 

 influence on S,- and ;S'/. As S,. and Si are independent of the choice 

 of a system of axes, we make a suitable choice by selecting the 

 X^ axis along the axis of reflection. Let us call «j, a,^, «3, it^ the 

 cosines of direction of a vector before the reflection'; ^:?^, ii,, /i^, ji, 

 those after it ; let us moreover represent u.^ a^" — «/ a^" etc. by 

 1,3 etc. and ^.,' ^," — l^,' il," etc. by x-,» etc. and let us call X,, etc. 

 the coeflicients of ]>osition of a plane with sense of rotation included 

 before the reflection ajid ft.^3 etc. those after it. Then : 



«1 = ^1 



«2 = t?2 



«4 = — <?4 



. S33 '■2S *14 /l4 



S3I Xri !?24 — - X34 



*12 A12 5«34 X34 



»/5%3 + i%i+^%2-f^%4 + S%4+r34 = |/X%3+X%: + X%2+X%4+X%4 + X^4. 



So also : 



52 



Proceedings Uuyal Acad. AmsLeidam. Vul. VI. 



