077 



points ^4 iiiid /), for they belong eucli (o two triangles of in\(»liition, 

 of which one has a vertex on r, the other a vertex on s. 



As the singular curve {B)'" cuts each of the lines r, s in m points, 

 Q and a have an ?/^fold point in B. The numbers m must therefore 

 satisfy the relation {n-\-3y = {n-\-'3'j-\-2-\-a -\- ^tn^ or 



« + Xm' = (/i-f l)(n |4) (3) 



In a similar way we find the relation 



«+ Vu^ = (,* + !) (n + 4) (4) 



From the relations') (1), (2), (3), (4) ensues moreover 



^m- = :Vfi- (5) 



:^(m-iy ^ ^{(i-iy, , (6) 



consequently 



^(2m-l)=r V(2^-l) (7) 



and 



« + ^(27?i— l) = 5(n-i-l) (8) 



5. The points P', P", of which the connecting line p {)asses 

 through E, lie on a curve a*, which has a node in E, and is touched 

 there by the lines EE', EE". 



If JS" is a singular point B then this locus consists apparently 

 of {By^ and a curve of order (4 — m). Hence m may be four at 

 most. If m = 3, s^ degenerates into {By and a singular line. 



Through E, six tangents pass to e^ ; each of these lines bears a 

 coincidence of the involution [P^). Such a line belongs to a group 

 of the involution {jf), in which p" is connected with //, The 

 coincidences of (p^) envelop a curve 73 of class three, recipi'ocally 

 corresponding to the curve 7', which contains the coincidences of (/^^j. 



By complementary curve we shall understand the envelope of the 

 lines p, which form triplets with the coincidences of the (p*). From 

 what was stated above follows therefore, that the complementarj' 

 curve of the (p') is of the sixth class. 



Analogously we find a complemeniari/ curve of the sixth ordei-, 

 >:", as locus of the points P, which complete the coincidences of the 

 (P^) into triplets. It has nodes in all the singular points of (P'), 

 for each curve (P)'« and each line a bears two coincidences, whicli 

 form triplets with the corresponding singular points. 



As the curve (P)"' has an (m — Ij-fold point in B, the curve of 



A) In my paper "A quadruple involution in the plane'' (These Proceedings vol. 

 XIII, p. 82) I have considered a (P^). which possesses a singular point of tlie 

 fourth order and six singular points of' the second order. In correspondence to 

 the formulae mentioned above, w = 4 was found. 



