44 Proceedings of the Royal Irish Academy. 



defined by ^ = 0. (The specification of the point t involves of course the 

 choice of one of the values of t' corresponding to the given value of t, i.e. the 

 choice of the sheet of the surface on which the point lies.) 



The functions x, y, z will each have n zeros, and will therefore be functions 

 of order n. If the points at inlinity are distinct, and none of them is situated 

 in any of the coordinate planes, the three functions have the same poles, all 

 simple. We can suppose any plane to be the plane at infinity : in other 

 words, we can suppose, if necessary, that x, y, z, 1 are proportional to the 

 homogeneous coordinates of a point on the curve with respect to a finite 

 tetrahedron ; hence we can suppose the poles to be common to x, y, z and 

 simple. 



The plane coordinates will be given by three functions X, fi, v, rational on 

 the Riemann surface, and satisfying \x + /xy + vz = 1. 



The order of A, fi, and v will be iV, the class of the curve. Their poles 

 will give the osculating planes which pass through the origin. These we can 

 also suppose simple, and common to A, ju, v. 



20. Consider the function U = XtjXt, where xt and X( are the deter- 

 minants (xi/z) and (kjlv) (corresponding to D and A of the last section). 

 U is a rational function on the Eiemaun surface ^. 



(a) Near an ordinary point 3f (t = <„) on the surface, which is not a pole 

 of (x) or of (A), we have, writing if for t - t„, 



T, y, z, A, fx,v = a + U + cf^ + . . . 



Hence Xt and A^ are finite or zero. The point will be a zero of xt of 

 order 2^ + q + r - 6 = 2p - Q, provided that we can write by a change of 

 coordinates 



X = afP + a'tP^' + . . ., Y=Ut + ..., Z^cf +...., {'P>q>r,p>?j), 



X, Y, Z being hnear (integral) functions of x, y, z, 1. 



When this is so, the osculating plane of the curve will be 



AX^BY^ CZ+ D = Q, 

 where we may take A, B, G, D ec[ual (and not merely proportional) to linear 

 functions of A, ^, v, 1. Then, as shown in section in, the orders ia t of D/A, 

 C/A, B/A are respectively p, q, r (since we are dealing with a P-eurve). 

 Again, A is not infinite at M, since the point is not one of the poles of 

 X, fx, V, 1 ; nor is it zero, siuce in that case A, B, G, D would be simul- 

 taneously zero, and the determinant of transformation from [Xfivl) to 

 {ABGD) would vanish. We can therefore write 



^ =«! + ..., B = b,t'- + ..., G - -.ti + . . ., JD = d,tP + . . 



