Egan — Linear Complex, and a certain class of Twisted Curves. 45 



Hence the determinant At of which one row is \ A A A A \ has a zero 

 of order 22:1 - 6. Now At/Xt = const. ; hence in the neighbourhood of such 

 a point U = .rt/Xt is finite and regular. 



(b) At a pole of .-», ?/, z, xt has a pole of order 4 ; X( is finite, since the 

 point is an ordinary point on the curve and 2p - Q =0; hence U has a pole 

 of order 4. At a pole of A, ^, v, V will have a zero of order 4. 



(c) Near a branch-point on the surface, we can write 



t -t, = c,'^, t' - t'„ = li^"^ + hc,^'' + . . . 



and hence by a change of coordinates, 



X=aZp+... Y=bZ'> + ... Z=cZ.'-+... 



It follows that as in («), a-^/X^ is finite, where the determinants .r^, Xf are 

 formed by analogy with a-^ Xt. 



Now ^ ^^ = *■*(!/' ^f = ^'(|J' 



hence U is finite. 



(d) We can deal with t = co hj writing r = t'\ Hence (whether the 

 surface has or has not a branch-point at infinity) we find that, for the points 

 for which i = 00, the function Xr/Xr = xtjXt = U is finite. We assume that 

 none of these points is a pole of .r, y, z or of X, fi, r. This involves no loss 

 of generality, since we can transform t homographically. 



It follows from the preceding discussion that U is rational on the 

 surface, has the n poles of a-, y, z as poles of order 4, the N poles of A, fi, v as 

 zeros of order 4, and has no other poles or zeros. 



21. Since U must have the same number of poles as of zeros, n = N. 

 The degree of an algebraic P'-curve is equal to its class. 



Again U is the fourth power of a rational function of the n\h order. For 

 let the four determinations of Ui corresponding to a point M on the surface 

 be w, iio, - w, - iw, = lo, w^, ivi, w^. It is easily verified that if M describes 

 any closed path on the surface, each of the four functions w, w,, ^v■i, w^ 

 returns to its original value. Any one of them, say w, is therefore uniform, 

 and has no singularities on the surface except n poles. Hence it is a rational 

 function of order n. 



22. Now if we write 



Xi = X, Xi = y, X3 = z, Xi = 1, 



tti = Xiv, 02 = fiw, as = vw, III = w, 



the functions Xi and a, will have the same poles (except Xi, which is constant) 

 and neither set will have a common zero. 



