( -^-l ) 



Mathematics. ■— "A yitrllculur si-j'ics a f qundnitic .^itrfdces irith 

 eiijht coiuiHon i>ouits nad c/if/it cunnnoii Imuji'iitidl planes." 

 By Prof. P. H. ScirouTE. 



1. "In our space are given a fixed line and four projeetively 

 related plane pencils of rays. To ite found the common transversals 

 of the fixed line and a set of four corresponding rays." 



Notation '). We indicate the fixed line by /", the vertices and 

 planes of fiie pencils of rays by 0^,0^,0^,0^ and «,,«,,«3,0^, 

 four corresponding rays and their two transversals by ?^, I„ /,, /, 

 and /, /', the pencils of rays themselves by (/,), (/,), (/,), (/,) and the 

 pairs of points of intersection of /, /' with each of the rays /,, /,, /,, /^ by 

 {S„ S\), iS,, S\), {S„ S\), {S„ S\). Farther the symbols /„„ /,„ .... I,„ 

 may represent the lines of intersection of the pairs of planes 



(«1. «,), («1. O3)- («3. «J- 



2. The order of the locus of the pair of transversals /, /' is easy 

 to deduct from its section with «j, which consists of two parts : the 

 locus [{Si, aS/)] of the pair of points (<Si, 5/) and some generating 

 transversals. Each ray /, of the pencil (/j) containing a single pair 

 of points (5i, jS'j'), the locus [{S^, *?/)] is an hyperelliptic curve the 

 order of which exceeds the number of times a transversal passes 

 through (>i by tsvo. Now three transversals pass through (1),. By pro- 

 jecting the pencils (/,), (/j) out of (>, on «^ we find namely in «^ three 

 projeetively related pencils (/'J, I\), (/J and now three times three cor- 

 responding rays /'„ l\, I^ pass through one and the same point, the 

 conies generated by the pairs [f/'J, (/j] and [(/',), (/j] having besides 

 O^ three more points in common. So the locus [{S^, S\)] is a curve 

 c'l' of order five having in 0^ a threefold point ; its genus is three. 

 Now that three transversals pass through Oi there must be according 

 to the principle of duality also three generating transversals in «j. 

 And indeed, the pencils (/j), (/b), (0 tlo describe on the lines Z,,,, /j,,, 

 /,,,, three projeetively related series of points {A,),{A^),{A^) where 

 three times three cori'esponding points A^, A^, A^ lie on the same 

 right line a, the conies generated by the pairs [{A^), {A J] and [(^4,), {A^)] 

 possessing three more common tangents besides /j.^. So the total section 

 of (t, with the locus of the pair of transversals /, /' is a system of 

 order eight and this locus itself a scroll 0" with a nodal curve of 

 order eighteen. The order of the nodal curve ensues even from the 

 fact that the surface C" must correspond in genus to c/ ; moreover 



1) For Udtution and reasoning seq a former communicnliüll. 



