( 61 ) 



6^ is the angle between the radius vector and the north-axis at 

 noon, T is the moment at which (he radius vector has the direction 

 (( of the great axis or, whenever the eUipsc is tlattened down to 

 a straiglit line, the moment when it attains its maximum value. 



It appears from tiiese tables that tiie gradient ellipses, for both 

 stations and in all seasons, approach to a straight line, so that a 

 graphical representation could only be given on a large scale. 



It would not be difficult to protfer an explanation of the somewhat 

 startling result that the angle of deviation varies with the diiferent 

 seasons. Such an explanation could be based only on a premised 

 conception concerning the mechanical meaning of the friction coeffi- 

 cient, as introduced in the calculation, and would be |)remature 

 before the results obtained have been put to the test by a[)plication 

 of the method indicated in this paper to other series of observations 

 made at manv and differentlv situated stations. 



Mathematics. — ''The pentagonal projections of tJie regular jlreccll 

 and Its seniiregular ojf'spring." Communicated by Prof. Scholte. 



1. Fundamental theorem. If in two circles (fig. 1) with radius q 

 situated in the planes 0{X^XJ , 0{X^X^) of a rectangular system 

 of coordinates in space aS^ we describe two regular pentagons 

 (1, 2, 3, 4, 5) , (1', 2', 3', 4', 5'), of which the first is convex while the 

 other is star shaped, the five points F^, P^, .., P5, whose projections 

 are the vertices of these pentagons indicated by corresponding num- 

 bers, form the vertices of a regular fivecell with Q\/b as length 

 of edge. ^) 



1) This theorem is not new. Probably it was given ibr the lirsl lime by 

 Dr. S. L. VAN Oss in his dissertation (Utrecht, 18941 Compare also my paper: 

 "Les projections régulières des polytopes réguliers" (Archives Tei/ler, Haarlem, 1904). 



We repeat here the simple proof. If (Pjo, P34) and (Qi^, Qu) ai"e the projections 

 of the points P and Q with the coordinates x; and /ji {i—-l,2,S, 4) on tiie planes 

 OiX^Xi) , 0{A\X^), we have 



and therefore if d denotes the distance PQ 



Now the projections /^i2^t^i2 ^^^^^ /^:uV:5i of each ot the ten edges lii, ..., 45 of 

 the fivepoint PiP.2P-P4,P:, are either side and diagonal or diagonal and side ot the 

 same regular pentagon, etc. 



Which position has the regular simplex .b\5) witli respect to the planes of pro- 

 jection O(XiXo) and OtA'sA't)? Evidently this projection is characterized by the 

 fact that each of the five pairs of non intersecting edges 



(25) (34) , C13)(45) , (iJ4) (15) , (35) (12) , (14) (23) 



