Egan — Linear Complex, and a certain class of Twisted Curves. 39 



For a P-curve the two numbers p + q + r ~& and op -q-r -Q are equal 

 at every point to 2p - 6. It follows that B and A have the same roots, and 

 therefore A/-D = k, where fc is a constant. 



Conversely, \i B = kA, our reasoning shows that p = q + r at every point ; 

 the property is therefore peculiar to P- curves. 



10. If we choose the parameter t so that the point ^ = oo is an ordinary 

 point on the curve, the degrees of B and A are respectively 4w - 12 and 

 4iV- 12. Hence the class of a rational P-curve is equal to its degree. 



Again, we have 4w - 12 = 2'2(p - S); (9) 



and hence, to generalise the theorem {due to M. Picard) that a rational curve 

 belonging to a linear complex has 2n - 6 inflexions, vje must count a (p, q, r) 

 point as equivalent to p - 3 inflexions. 



11. For a P-eurve, every root of B (or A) is of even multiplicity {2p - 6); 



hence B is the square of a polynomial of order 2n - 6. To find this polynomial, 



we write 



^ d">a! d"Xi , , 



Multiplying the determinants B and A by the ordinary rule, and 

 remembering that, since (a) is the osculating plane at (a;), 



(00) = (01) = (10) = (02) = (11) = (20) = 0, . . . (10) 



we find 



IB" ^BA^ (03) (12) (21) (30). 



If we differentiate the last three of the equations (10), we find 



(03) = -(12) = (21) = -(30) (11) 



Hence 



hB^ = (03)^ . . . (12) 



The required polynomial is therefore (03), aside from a constant factor. 

 The identity (12) is obviously equivalent to kB = A, and is therefore peculiar 



to P-curves. 







12. Again, we have 







Ol 



02 SajiCj 



(03) 



dBjd^, 



dBldx. ■ ■ ■ B 



B 





(04) 



(04) 





1 XiXiiCiXi I 



i) 



Hence 



i) (04) 

 B (03) 





In like manner 



A (40) - (40) 

 A (30) (03) 





