60 Proceedings of the Royal Irish Academy. 



Besides the ordinary (3, 2, 1) points which annul it, (33) may vanish at 

 some of the singularities. In effect, if we use the equations (5a) and (6) of 

 section iii to determine the initial coefficients of the functions m in the 

 neighbourhood of such a point, we can write 



Xi = af + . . ., a, = hcdqr (r -§•) + ... , 



Xi = W + . . ., 02 = cadrp {p - r]t^ + . . ., 



Xs = cf ■\- . . ., 03 = abdpq {q - p) t^ + . . ., 



Xi = d ■¥ . . ., at = abc {p - q) iq - r) [r - p)tP + . . ., 



(p = q + r). 



Suppose /) > 5 : then the term of lowest degree in Sa'jii = (33) will 

 be of degree p - 5. li r< 3, this is evident ; if not, the coefficient of f-''- 

 is derived from azi's + 03^3 and is 



ahcdq lr][rp {p -r) +pq{q- p)]/{q - 2) ! (r - 2) ! 2 ! 2 ! 



Since p - r = q and q - p = - r, this vanishes. Hence a singularity 

 for which p> 5 is a root of (33) of order p - 5 (or in particular cases of 

 higher order). 



In the case of an undulation (5, 4, 1], it is easily verified from the 

 expressions just given for the coordinates that (33) has a root at the 

 undulation. 



51. From what precedes we can infer that if a sextic P-cwve [rational or 

 not) has a point for which p = Q, or an und^dation, or an ordinary (3, 2, 1) 

 point at which six consecutive tangeTds belong to a linear complex, all the tangents 

 to the curve belong to a linear complex. 



In effect, if we write 



W,{t, d) = tai{e)xi{t) + ^a,{t)xi(d) = (0 - tyTF(t, 0), 



and if t„ be the special point just referred to, then the function TF,(!!j„ 0) of 

 has <o as a zero of order seven, since 



Lt w,{t„ ey(e - t,y = - 2 (33)j6 1 = 0. 



But Wi {t„, 6) is of the sixth order in 9, and therefore can only have a zero 

 of order seven by vanishing identically. 



Now consider the function Wi (t, d) where t ^ t^,. It has the zero 6 = t 

 of order six and also the zero 6 - t„: hence it vanishes for all values of 

 and t : and this requires that the curve should belong to a linear complex. 



In the case of a rational P-curve the function (33) is of order 2?i - 12 

 in t, since it gives the coincidences of an ?i - 6, n - & correspondence. 

 On an algebraic P-curve its order is 2n - 12 + 2jd, where y is an integer 

 and d is the deficiency. 



