( 783 ) 

 The particulars of this research will he published later on else^A^here. 

 Amsterdam, March J 904. (hy<(n. C/wm. Lab. of the Uii'ir. 



Mathematics. — "Rei/ular jn-ojectluns of rcijiilar jio/i/fo/n's.'' By 



Prof. r. H. SCHOUTE. 



We consider fur this end the three regiihir |>olyto[»es .1,,, />,,, C„ 

 of the space S„ with ii dimejisions, which correspond respectively 

 to the tetrahedron, the hexahedron and the octahedron of oui- 

 space and setting aside the [)olytope B,, willi its exceedingly simple 

 properties we treat some special cases of tlie following' two geneial 

 theorems relating to A„ and C», of which the proof will be given 

 elsewhere. 



Theorem 1. 



"Let m represent in or ^(?i-j-i) according to the nund)er of 

 dimensio]is ii of S„ being even or odd." 



"Construe in m planes «^, «^ . - . n,a congruent regular jiolygous 

 with n-\-l [or 2y/,] sides; let (> be the circumradius of those polygons." 



"Let us take in each of tliose planes a vertex of the polygon as 

 the origin and a definite seuse, in which distance is counted from 

 this origin to any other vertex along the circuit." 



"Let us place at the remaining vertices the numbers 1, 2 . . . 

 in such a way that the number j' i^ pid in ft/- near the vertex which 

 is distant from the vertex iu the sense assumed in «/, a number 

 pk [or /) {2k — J)] sides. In other words: let us place in <(i, moving 

 round from iu tiie indicated sense the iiiimbei-s 1. 2 iu such n 

 way that when contimdug to a following iiumhei- we skip /—I 

 [or 2 {k — 1)] vertices. Here the polygon in '</.■ can be rexhiced as far 



sides, 



as the numbering goes to a regular poKuon with 



7 

 each vertex of which bears </ niunbers, as soon as /• and //.-|-1 



or 

 '7 



53'^ belong:; to a cunditiou of equilibrium betwecii Ihc twu uxide I'oiuisj. The 

 questiou put lo lue by Messrs. Behkünd and ItoTU in their recent paper (Ann. 

 331, 359) has tlierel'ore now been answered. .My furnier contention that glucose 

 with |«|/y+l()6" might crystallise from a solutiun in which it was not present 

 (namely, from glucusu willi a Yx\i)-\-'ji'i^\ is, uf couisf, no lunger tenable. It is, 

 as B. anti K ubsurve, a queslion of the relative solubility of the two or three 

 isomers able to be cuuvcrlcd iulu cacli oilier. I Jiad already sliart'd Ibis view for 

 a cunsiderablf time. Lownv an<i Fn. Akmstko.m; have alst) ex|)ressed the opinion 

 that it is a question of u(iuilibrium. As staled above, Mr. JuNfiius will try lo delennine 

 the precise nature of Ibis equilibrium. L. u. B. 



