[ 33 ] 



III. 



THE LINEAE COMPLEX, AND A CERTAIN CLASS OF TWISTED 



CUEVES. 



By EEV. M. F. EG-AN, M.A., 

 Lecturer in Pure Mathematics at University College, Dublin. 



Read Febkuary 13. Published June 24, 1911. 

 CONTENTS. 







PAGE 







PAGF. 



I. 



Summary 



33 



VII. 



Metrical Results, 



51 



II. 



Generalities 



3.5 









III. 



Extension of a Theorem of 





vm. 



P-Quintios, .... 



64 





Picard, .... 



3.5 



IX. 



Asymptotic Lines of Ruled Sur- 





IV. 



Rational P- Curves, . 



38 





faces belonging to a Linear 





V. 



Algebraic P-Curves, 



43 





Congruence, 



65 



Yl. 



Sufficient Conditions that an 













Algebraic Curve should belong 





X. 



TheP-Quintic with aBitangent . 



69 





to a Linear Complex, . 



47 









I. — Summary. 



The claws of twisted curves with which this paper deals is characterized by 

 the property that the class of each cycle of the curve is equal to its degree. 

 These curves are called P-curves in the paper. 



In section in it is shown that every curve whose tangents belong to a 

 linear complex is a P-curve. 



Sections IV, v, vi, and vii are the result of an attempt to prove that all 

 algebraic P-curves belong by their tangents to a linear complex. This I have 

 found not to be the case. 



In section iv rational P-curves are discussed, the homogeneous point- 

 coordinates (x) and plane-coordinates (a) being represented by polynomials 

 in a parameter t. It is shown that the class of a P-curve is equal to its 

 degree, and that such a curve is characterized by one of a number of 

 equivalent identical equations. If we write 



T dt^ "' dF^ ""' ^ ^""'^' *-* ^ ^' ^' ^' ^^' 



one of these identities is 



(22) ^ 0. 



In section v these results are extended to algebraic P-curves. It is 



R.I.A. PROC, VOL. XXIX., SECT. A. [5] 



