293 



the intersec'lioiis X of p with [Py su|)|)ly t'lirtlier 4 common 

 points A'. The remaining JO points which they have moreover in 

 coinmoii form 5 pairs A', A'", of which the line of connection :c 

 passes through I\ In other words, if A' descril)es a straiglit line, .<.• 

 envelo[)S a rational cnrve of the ,/i/th class. 



3. Let us now consider the case that ihree base-points H,, B^, B^ 

 of a [c'] lie on a straight line a, while the remaining three. C\, 6',, 6',, 

 have been chosen arbitrarily. 



To the net belongs a [)encil, each cnrve, of which consists of the 

 straight line d and a conic Ihal passes through C'l, C,, (\ and a 

 certain point A. These conies determine an /, on a, the pairs of 

 which are completed by ^4 into groups of the /,. So A \s n sm(/ii.lar 

 point, a a singular straight line. 



To the singular points (\, t\, C\ the nodal curves y/," are again 

 associated as before; to the singular points /i,, />,, /J, now belong 

 curves /i'i, which pass through the points C aud .4. Eacii {i/^^ forms, 

 as is known, with a the net-curve that has a node in Bk- 



On the pair of lines AC^, C, C\, [c'] determines a system of 

 groups of the /,, a point of which lies every time on C\ C\, so that 

 AC^ contains an 7, of pairs A, A"'. The three straight lines ca, = -4Cjfc 

 are therefore singular, they form with the singular straight line a 

 the curve {Py of the point .4 (see § 2). 



For Ck.{Py consists of y^ ^^^^^ <-'k, for Bk of /?^% ^/ and a singular 

 straight line bf.. There are consequently seven singular points {A, Bk, Ck) 

 and seven singtilar straight lines {a, bj-, c/J. 



The straight line a is compouent pai-t of the Jacobian, the curve 

 of coincidences is now a y^ that passes through the three points B 

 and has nodes in the three points 6'. 



The curves {p)^ and {q)^ have now oidy 7 tangents .r', in common, 

 which connect a point A of p with a point A' of q. In connection 

 with this />"* is now replaced by a //. which passes three times 

 through Ck, twice through Bk. 



Between the points A' of j) and the points A^*, which ai-e every 

 time produced by the intei"section of .r on y>, a correspondence exists, 

 each coincidence of which is at the same lime a coincidence of the 

 /, ; hence x envelops a curve of the fourth class, when A' describes 

 a straight line. 



4. Let us now supjiose that one of the six base-points of [c*] is 

 collinear with the base-points B^, B*, and with the base-points 



