( 579 ) 
two conics of the locus on account of its cutting c; twice; so S$ breaks 
up into this double plane and an S® with fourfold line / and fivefold 
point S=le;. 
This surface S® contains the common transversals azim, bkim of 
l, CkyCly Cm, completed to pairs of lines by the lines bn, an through S 
meeting ec, Moreover it passes through the lines az through $ 
meeting cx and ci, completed by the lines ban cutting eme. So the 
14 pairs of lines situated in S® (§ 1) have been indicated. 
If S® still admits of a fourfold line / and four simple lines cj, ca, ¢g, c4 
crossing it, but not of a fivefold point S, the 14 pairs of lines are 
enumerated as follows. Firstly we indicate the 8 transversals az1m, brtm 
and the corresponding lines bn, an. By remarking that the quadric 
determined by J, ¢,¢, cuts S® still in an R breaking up into six lines, 
and that aj93, bios, G24) 4j94 are amongst these, it is found that the 
missing pairs of lines can be denoted as az, bmn. 
§ 5. The conics through the given points S,S' meeting the 
lines ej,ez,ez generate an St with double line /=$$' and triple 
points S,S', derived from the S* by supposing that c, and cs meet / 
in S and S. Of the eight pairs of lines cutting / six can be 
represented as az, bm and bt, am, the lines a of which pass through 
S and the lines 5 through S'; the missing pairs are formed by Zand 
one of the common babes of / and the lines cx. 
By the intersection of this particular surface S* by a twisted 
curve Rp meeting Za number of (p—1) times it is found that the 
conics through S,S' and meeting R?, ej, cz generate an S*2*? with 
two (2p+1)-fold points S,S’ and a 2p-fold line J. The 2p lines 
forming with / degenerated conics of the system can be indicated 
as follows: the quadrie (ley cy) cuts Rr in still (»+1) more points, 
each of which gives rise to a common transversal of J, cj,c and Rp 
lying on S%+2; the remaining (p—1) transversals of J, ¢), eo, Re lie 
in the planes through 1 touching Rp in its points situated on J. 
The cone (S, Re) determines with ¢; (and with co) p transversals 
of Rp and cj (cs). The same holds good for the cone (S', Ap). Adding 
to these 2p transversals the lines through S (and S') cutting ¢ and 
co, we get the (4p+2) pairs of lines, forming with the 2p pairs of 
lines amongst which figures / the (6p+2) pairs of lines S*e+? must 
contain. 
§ 6. If a surface S ++! contains a w-fold line m and a v-fold 
line n crossing it, then the skew surface generated by the lines 
meeting m,n and any given plane section C#+'+1 of S#+’4! admits 
