(13) 



Egan — Linear Complex^ and a certain class of Twisted Curves. 4l 



14. To summarize the properties we have shown to belong to a rational 

 P- curve : — 



(1) The class is equcd to the degree. 



(2) S (2? - 3) = 2«, - 6. 



(3) The curve is characterized lij the following identities, any one of ivhich 

 involves the others : — 



(a) p- q -r^O; 

 (h) A = kl>; 



(c) kW = (03y; 



(d) (22) = 0, from which follows 



(e) {mn) + {nm) = 0, (m + n < &); 



if) ft = AujXij = const. 



Note. — We shall see in the next section that, for an algebraic P-curve, we can represent the 

 coordinates Xi and m by functions of order n, rational on a Eiemann surface, and having their poles 

 in common ; and for these the ec[uations (13) will hold. 



Note on the General PMtional Curve. 



15. For any rational curve we can write 



IX = Suij Au/^dij JTij = AjX, 



where the (lij are arbitrary constants. It is easily verified from equations (7) 

 and (8) that, at a point witlr the characteristic numbers p, q, r, A and X Irave 

 zeros of order r-1 and p -q-1 respectively. The other zeros of X are the 

 points where the tangent belongs to the complex laijpij = 0, and these are 

 also zeros of A. Eemoving these, we find 



fi = hu{t- t,y-'in (t - tsY-i-' = h n(t- ^,)««-^ ... (a) 



where the continued product applies to all points ts at which either r or 2^-q 



is greater than unity. 

 Again, 



B/A = hn{t- t,)p^i^'-<^-i'P-i-'-'> = h, U(t- tsf^i''-^^ = h K- ... (13) 

 Hence hfi' = ^V^^ = ^V(03)S fi = A;,2)/(03)l ... (y) 



A and 2' are respectively of degrees 2iV" - 2 and 2n - 2 in t, iVand n 



being the class and degree of the curve. The order of /x for ?: ^^ oo is therefore 



2(N- n). Hence 



^(q + r-p) = 2(M-n). ... (8) 



Again, (03)' = i)A = h U(t - t,y"-'\ 



hence (OS) = hn{t - t,y-\ . . . (e) 



R.I.A. PROC, VOL. XXIX., SECT. A. [6] 



