300 



= 



in which Ox^ etc. are functions of ^'j and .f^. All cMiave a node in yS^. 

 The groups of the /^ on the rajs passing through S^ consist of 

 the point <S'„ twice to be counted and a point of the curve o\ 

 indicated bj 



= 0, 



which lias a triple point in »S'o. 



As (Pj* has a node in the singuhir point *S'j, /'* bears 6%A^ straight 

 lines /, so that t is now of class 8. The curve of coincidence y' 

 intersects (/'*)'' in the points of contact of the 8 straight lines t and 

 twice in each of the 9 simjular points Sk (single base-points of [c'']) ; 

 fi'om this it ensues that y' passes five times through *S'o. 



We now consider two arbitrary pencils of the net [c"*] and associate 

 to each c* of a peiu'il the cur\es of the other, which curves intersect 

 it on y". The product of the pencils tliat consequently are in an (8,8) 

 consists of the twice counted curve y', eight times the c\ which 

 the pencils have in common, and the roinpleiiientary curve .r. From 

 64 — 2x9 — 8 X 4 = 14 it now appears thai ,/■ is a curve of order 14. 



The curve o^k belonging to *S;fc lias nodes in >S';t fiiid -.S,, ; consequently 

 is Sjc qiuulruple point of ,<■. A combination of {Py with a'^" now leads 

 to the conclusion that .v^* possesses a se.vtuple j)oini in .S',. 



We now find by the combination of y* and ,u^^ that 7, contains 

 12 groups in luhich the three points luive coincided. 



The characteristic numbers of r are easy to find, as this curve 

 corresponds in genus to y', and has the 12 points of contact of y 

 and .r as points of inflexion. It appears to be of order 20. 



12. If in 



= 



05.1* etc. again represent functions of x^ and a\, all the curves of 

 [c"*] in 0^ = Sq have a triple point. The groups of the /, are now 

 determined by 



kdr^ + //>./ + rncx^ = and kx^ -\- /a-, -f- ?»'^» = 0. 



The first of these equations shows that the rays have been arranged 

 by ,S, into the triplets of an involution of the second rank. 



