Mathematics. — '*A relation betiveen the po/yiopes of the Cmo- 

 fauiily". By Dr. W. A. Wijthoff. (Comtniiiiicafed by Prof. 

 J. Cardinaal). 



(Communicated in the meeting of June 30, 1917). 



1. In the manuscript left by the late Prof. Dr. P. H. Schoute 

 which served as basis for the fifth section of his last memoir on 

 polytopes ^), we find with reference to the form of the coordinate- 

 symbols of the polytope ce^ e^ Cgoo ^^le following remark : ''Result 

 much resemblance to ce^ e,." 



The resemblance consists not only in the total number of vertices 

 being the same, but also in the fact that this number is distributed 

 over the same number of symbols which agree with each other 

 in the number of zeros and in the number of vertices represented 

 by each symbol. 



Again, in the same manuscript the author makes this remark 

 about the coordinate-symbols of cei e, C\ot> '• ''Much resemblance to 

 Elte p. 26, but not the same. Ha.^ to be farther investigated." ') 



The time for makmg this further investigation, however, was lacking, 

 neither did the writer observe a similar resemblance which exists 

 between several other members of the said family as also between 

 members of other families. 



In this paper I intend to explain the existence of these resemblances. 



2. Let each of the 600 regular tetrahedra, the limiting bodies 

 of the C'eooj be divided into 24 other tetrahedra by the 6 planes 

 each going through one. of the edges and the middle point of the 

 opposite edge. Project these planes from the centre on the circum- 

 scribed hypersphere. 



1) P. H. Schoute. Analytical treatment of the polytopes regularly derived from 

 the regular polytopes (Section V). Verh. der Kon. Akad. v. W. te Amsterdam. 

 Eerste sectie, Deel XII, N". 2. 



J. Gardinaal. Mededeeling over een nagelaten arbeid van wijlen Prof. P. H. Schoute,' 

 Verslagen, Deel XXIV 2, p. 1077—1079; These Proceedings XVIII p. 1173—74. 



2) This remark refers to the dissertation of Dr. E. L. Elte: "The semiregular 

 polytopes of the hyperspaces." But the page is incorrectly quoted, and should be 

 p. 25. See the footnote to § 7. 



