( 489 ) 



Further follows from (15), (17) and (18): 



1 

 n k con a z=: o"^ Q^ .sin 2 (t -f- (o). 



According- to (2) the second incnd)ei- is e(inal to a" .shi 2t and so 

 according- to (3) to n^ /:„, and thns tlie second e(|nation has also 

 been derived. 



To conclude we may remark, that here the I'eversed course has 

 heen taken from that hv whicii in the j)receding paper the occur- 

 rence of the so-called com})lex index of refi-aclion was derived from 

 the two fundamental equations ^). 



Mathematics. — ''A tortaoas svrfacc. of ordi'i- si.v (ukJ of (jenus 

 zero ill space >S/>, if four duiieiisloiis.^^ Hy Prof. 1*. H. Schoute. 



'1. We begin bv putting the following question: 



"In space Sp^ are given three planes n^, a.^, <i^ and in these are 

 ''assumed three i)rojectively related pencils of ra\'s. We demand the 

 "locus of the common transversal of the triplets of rays corresponding 

 "to each other." 



JSfotatlon. We indicate the vertices of the rays of pencils by 

 (>i, (>.,, O^, three corresponding rays and their transversal by l^, 4, /, 

 and /, tlie points of iiitersection of / and 1^,1.^,1^ hy S^, S.^, S^ and 

 the pencils of rays by (/J, (4), (/j). Let fnrther l\^^, P^^, P^.^ indicate 

 the points of intersection of the planes «^, a^, ct^ two l»y two, and (c 

 the plane P.,^ P^^ P^, which has a line in common with each of 

 the i)lanes tc^, a.^, «.,, namely with «^ the line P^^ P^^ = (ii, vvith «.^ the 

 line l\^^ P.^^ z=: (i^, with a^ the line P^^ P^^ ■= a^. We take for 

 granted that not one of the three vertices O-^, 0^, (.>, coincides with 

 one of the points P.^^, Pj,, P^^. 



2. The answering of the given question oilers no more difliculties, 

 as soon as the locus of point >S'i in «^ is known; so we shall first 

 lind this. Each ray /^ of pencil (/J furnishing a single point ^S\, it 

 is a rational curve, wliose degree surpasses the number of times a 

 transversal / passes through O^ with unity, ü^ow two transversals 

 / pass through 0^. For the pencil of planes {()^ /.J with ((>, «J as 

 bearing space and f^ (K^ as axis marks on the line of intersection 

 in of ((>i «J with «, a series of points (/'') projectively related to 

 the pencil of rays (/,), from which ensues that there are two rays /., 

 passing through their corresponding j)oint /■* and that therefore there 



1) See loc. cit. § 5. 



