64 Proceedings of the Royal Irish Academy. 



Hence F{t, 0) defines a 2?i - 9, 2n-9 correspondence between t and d. 

 J^ow the equation j (t, 6) = obviously means that a linear complex contains 

 three consecutive tangents at t and three at 6. It follows (pars. 48, 49) that 

 F = VW. Hence 



j(t, 9)^(0- ty B (0 D (6) VW. (26) 



Again, J= Lt. j(t,e)l{Q -if = DKt){VW)e-_t. 



Now when 6 = t, V and W, as we saw in par. 50, become (03) and (33). Hence 



J = (03) (33) D' = (03)^ (33), (27) 



omitting a numerical factor. 



VIII.— P-QUINTICS. 



55. Snyder has shown (American Journal of Mathematics, 1907,pp.279sq.) 

 that a qiiintic belonging to a linear complex is rational. As every P-quintic 

 belongs to a complex, it likewise is rational. 



56. A rational P-quintie (which we shall denote by the sjTnbol P5) cannot 

 have a double point with two distinct tangents. For if it has, take the two 

 parameters of the point as ^ = 0, t= x. Let Xiii;2X3 be the point : the equations 

 of the curve can be written in the form 



zj(at* + hf) = Xij{ct^ + elf) - .tHliet"- ^ ft) = .n/^, 



where 7 is a quintic in t. The osculating planes of the two branches are 

 therefore x^ = and 2-3 = ; these are cUstinct, and cannot both be the polar 

 plane of the point with respect to a linear complex. Hence the curve does not 

 belong to a linear complex, and therefore (since it is of the iifth degree) it 

 cannot be a P-curve. 



57. The only possible singularities are therefore the cusp (5, 3, 2), the 

 undulation (5, 4, 1), and the inflexion (4, 3, 1). Since 2 (7' - 3) = 2w - 6 = 4 

 (section iv, par. 14), we may have 



(a) two cusps, 



(b) two imdulations, 



(c) one cusp and two iniiexions, 



(d) one undulation and two inflexions, 



(e) four inflexions.* 



No quintic can have an undiilation and a cusp (see iv, par. 18). 



* When thia paper was read to the Eoyal Irish Acailemy, I had not seen Professor Snyder's 

 paper above referred to, in which this classihcation is reached by a different method. 



