265, 
just now, and the consequence of it is that a far greater number 
of intersections of the rest nodal curve and x, have coincided in 7, 
than we mentioned just now. 
We have to go back to the 3 different kinds of branches of the 
rest nodal curve passing through 7, which we summed up in $ 6. 
The branches of the first kind, arising from the sheets of 2 passing 
through the u—2e— 25 double generatrices going to the inte r- 
sections S, of k”and/ , appear in groups of 4 and touch in 
Z,, at the straight lines going to the intersections of the asymptotes 
of 4”; such a branch touches at the-4 sheets of a, passing through 
the same double generatrices as the sheets of @ that produce the 
group of 4, and has therefore in consequence of this already 8 points 
in common with zr. It further cuts the 2 (ua—2e—2o—2) remaining 
sheets of a, of the same kind one by one simply and also the 20 
sheets of +, passing through the 6 double generatrices of 2 that go 
io thee points. of ‘comtact=h, cof jk" and v,; 80: that it has 
totally in 74, 8-+ 2 (w—2e—2o—2)-+ 20 points in common with 
a. The group now consists of 4 of those branches and the number 
of groups amounts to $ (u—2e—2o) (u—2e—2o—-1); we find there- 
fore a number of points: 
a= 4842 (u—2e—26 —2) 4-20). § (u—2e— 20) (u--2e— 20—1). 
The second group arose from the sheets of 2 passing through 
the lines Z, A, which sheets all touch at ¢, ; the branches meant 
here appear in groups of 8. One branch out of such a group touches 
at all the 25 sheets of a, passing through the lines 7, 2. and has 
alone from this source therefore already 46 points in common with 
a, It cuts on the contrary the 2 (u—2s—2o) sheets of a, passing 
through the lines ZS, and the number of groups amounts to 
46(6—1); as a second contribution to the number looked for by 
us we find therefore: 
b= 8 {40 + 2 (u — 2e — 20}. § o(o — 1). 
Finally we have the branches of the 3'¢ group procured by the 
intersection of the sheets of 2 passing through the «—2s«—2o straight 
lines Z S, and the o straight lines ZR, ; these branches appear 
in groups of 4, and all touch at ¢,. Such a branch now touches 
in the first place at the 25 sheets of a, passing through the lines 
ZR, which therefore produces 46 points, touches then moreover 
at the 2 sheets of a, passing through the line Z,S,, which so to 
say belongs to the group, which procures 4 further points, and cuts 
the 2(u—2s—2o—1) sheets of zr, passing through the remaining 
lines ZS; the number of groups is moreover (u—28—20)0, so 
that the third and last contribution to the numbers wanted 
