34 Proceedings of the Royal Irish Academy. 



shown that the class of the curve is equal to the degree, and that the coordi- 

 nates can be represented bv two sets of foui' "rational" functions of the 

 analytic point t on. & Eiemann surface, all the functions having the same 

 poles, and neither set of four ha-\-ing a common zero : these functions satisfy 

 the identities of the preceding section. 



In section Ti it is shown that the necessary and sufficient condition that 

 an algebraic P-curve should' belong to a linear complex can be written in 



either of the equivalent forms 



(33) - 0, 



W, = 2a,(^).r.(0 + 2a,(0s-,(e) - 

 (for all positions of the points t and B on the curve). 



Two corollaries foUow. (1) An algebraic curve, which is such that the 

 osculating plane at a point A will meet the cm-ve again at 5 if the osculating 

 plane at B passes through A, belongs to a linear complex. (2) Every P-curve 

 of the fifth or lower degree belongs to a linear complex. 



Section vn is chiefly metrical. Considerable use is made of two theorems 

 on curves of a linear complex commimicated to the wi-iter by Professor 

 M'Weeney of University CoUege, Dublin. From one of these an expression 

 is deduced for the curvature of a complex curve in terms of the torsion and 

 its first two differential coefficients. By means of this the equations are 

 found of the most general cylindrical helix belonging to a linear complex. 

 From the second the value of the torsion of an algebraic P-cui'vo is deduced 

 in terms of the parameter t used in the preceding sections. By means of this 

 expression the conditions (33) = 0, Wi = for a complex curve are inter- 

 preted geometrically. It is found that a P-sextic will belong to a linear 

 complex if it contains a point of one of three given species. 



In section Tin the higher singularities of P-quintics are discussed. In 

 section ix some of PittareUi's results on the asymptotic lines of ruled 

 surfaces of a linear complex are given, with some additions. These lead to 

 the discussion in section x of the properties of the P-quintic with a bitangent. 



It may not be out of place to point out that the letter chosen to represent 

 this special class of curves is the initial of M. Picard's name. These curves 

 are characterized by what we may call the generalized Picard theorem (see 

 section m) ; they owe M. Picard a debt, and can claim a relationship with 

 him. 



