Egan — Linear Complex, and a certain class of Twisicd Curves. 43 



17. It is natural to ask whethei' to eveiy syuimetric polynomial V{t,ti), 

 of degree n-S in and in t, there corresponds a rational complex curve such 

 that F = is the condition that the line tO should belong to the complex. 



From what we have just proved, this is not so unless V satisfies the 

 following condition : — Let t and 9 denote the Cartesian coordinates of a point 

 in a plane. Then at any point (to, to) where the curve F = meets its axis 

 of symmetry t = 0, the three lines t = to, t = 0, = t^ must have contact of 

 the same order with the curve V. 



18. This condition is necessary but not sufficient. Consider, for instance, 

 the 2, 2 relation 



V=t'' + e- + ate = 0, (a + 2 + 0), 



and suppose that there is a corresponding rational qxiintic Q. The stationary 

 points* on Q are given by the roots of the qnartic, 



V {t, t) => {a +2) f= 0, 

 which has t = and i! = oo as double roots. Hence, at the points and co on 

 Q,p = 5. Each of these points is therefore either an xmdulation (p = 5, ^ = 4, 

 r = 1) or a cusp (5, 3, 2). Since any plane through the tangent at an 

 undulation meets the curve at only one other point, there cannot be a cusp 

 and an undulation on the same quintic. Hence the coordinates of a point on 

 Q may be written either t\ t\ t, 1 (two undulations) or t^, t^, f, 1 (two 

 cusps). The 2, 2 relations are easily found to be respectively 

 3;;' + 4t9 + 39' = 0, t' + MB + 0= = 0. 



Hence, unless « = 3 or 4/3, there is no curve Q corresponding to V. 

 I have not been able to determine the other conditions which V must satisfy. 



In the case of a P-curve not belonging to a complex, there is an 

 n - 0,11 - 3 relation given by V =0, where 



{6 - ty V[t, 6) = F, {t, 6} = Sa; {d) Xi (t) - Sai (0 X, (9). 



This relation is discussed in section vii, 48, 49, along with the n-6, n- 6 

 relation IF = 0, where 



{9 - ty W{t, 6) = T'F = %ai{e)Xi{t) + •Zai(t)Xi{9). 



V. — Algebraic P- Curves. 



19. Taking Cartesian coordinates, the equations of an algebraic curve of 

 the rath degree can be written in the form 



x = F,[t,t'), y = F,[t,t'), z = F,(t,t'), 

 where t and t' are connected by the algebraic equation ^ {t, f) = 0, and 

 Fi, Fi, F, are rational functions of the point t on the Eiemann surface 



* We use this expression to denote cycles of the curve for wliiuli p > Z. 



[6*] 



