985 



of the involutions {P, F') lying on tiio conies {BkY will analogously 

 be collinear. 



The line on which they lie contains four pairs {P', P"), wiiich 

 form each a triangle of involution with one of the points Bk; from 

 this we conclude that it is the singular line of 3'-^ order, which (^;') 

 must have. 



14. Let now n = 4. As to a line r a o~ must correspond, no 

 singular point of the 3'<^ order iS^^^ can occur beside a singular 

 point of the 4^^ order S^^^ (see ^ 13). A simple investigation shows 

 that only two cases are possible, viz. {i) one point *SW with sLv 

 points *S<^2) or (2) three points S^^\ with t/iire points S^~^ and one 

 point >S(i). 



The first case appears on further investigation to be realised 

 by the (P') mentioned at the beginning of § 12 ^) To the singular 

 point of the 4'^' order, E, belongs a rational curve {.Ey, which 

 passes also through the remaining singular points Fk, Gk {k ^= 1, 2, 3). 

 Singular lines of the 2"^^ order are FjcFi and GjcGr, the axes of 

 involutions (p', p") belonging to them we find in EFm iiud EG,n. 



As these six axes meet in E, the singular line of 4'^^ order will 

 contain the centres of the involutions /^ on the conies {Fkf,{Gi)'^. 



In the second case there are three singular points Ci!<^\ three points 

 Bk^~\ one point A, and, analogously, three lines C]l^\ three lines 

 5^•(-\ one line a. 



With the curve of coincidences y\ which possesses nodes in 

 Ck, (Ci)' has in common the 2 coincidences of the P lying on it, 

 and six points in 6\ ; the remaining 7 points of intersection must lie 

 in singular points, consequently [C^Y passes also through C^, C^, and Bk- 



On (Gt)^ lies therefore a point P, which forms a A with Ck and 

 B^; hence {B^Y passes througji Ck- 



The line a is transformed by (P, P') into itself and a figure of 

 the G"^'! order, so, either into the three conies [Bk)^ or into two 

 curves {CkY- Rut the second supposition is to be cancelled, because 

 a would contain 6 coincidences in that case, two of its /* and four 

 in the two points C. Consequently the points B^,B.^,B^ lie oji the 

 singular line a. 



Analogously the singular lines h^, h^. h^ meet in A. 



Every singular line c^• passes through a point Ck and completes 

 [CkY into a h\ 



The curve of the 3'^* class {c^)^ belonging to c^ has c.^, Cg, bk as 

 tangents (and c^ as bitangent). 



1) See also my paper, referred to above, in volume Xlll, p. 90, 91. 



