in relation to the theory of point-groups. 477 



Again, in the equation 2 <I>^IJ = suppose yjr to vanish at 



the points A. Then i/r contains /j, arbitrary coefficients and thus 

 the values of any at the points B are connected by /x linear 

 equations, of which none can be illusory since no i/r can vanish 

 at all the mn points A and B. Hence \, the number of arbitrary 

 coefficients in an n^° through the points B, 



^^(n+1) (n + 2) - mn + g + fjb, 



that is, fi-^X + mn -^(n+ l)(n -\- 2) —q (6). 



Comparing (5), (6), we have the theorem of Riemann and 

 Roch, that 



X — ^{n+l){n+ 2) — mn + q -^ /x. 



4. It is important to prove that no other linear relation except 

 (1) connects the values of an arbitrary (w + ?i— 8)*° at the mji 

 intersections of m = 0, w = 0, that is, that an (m + n — S)'*^ w can be 

 found such that 



W (Xr, 2/r) = O^r (^ = 2, 3 ... Viu) 



where the inn — 1 quantities a^ have any values whatever. Such 

 a polynomial is in fact given by Kronecker's method of inter- 

 polation, and is 



r=2 ^ \^ri yr) 



Hence no other linear relation than (1) connects the values of w 

 at the Tnn points, and from the course of the proof in § 1, any w 

 which vanishes at the mn points must be expressible in the form 

 u<^ + v\\r, where 0, -v/r are polynomials of the degrees ?i — 3, m — 3. 

 Two other proofs will now be given of this result, which was 

 first proved by Nother {Math. Ann. vol. 6, p. 354). 



5. In the theorem of Riemann and Roch put m + w — 3 for u 

 and suppose the points B to be {x^, ?/,■) (r = 2, 3 ... mn), so that q 

 takes the value m (m + n — 3) — (mn — 1) or m (w — 3) + 1. Thus 

 no (m — 3)'" can contain all the points A, and /u. takes the value 0, 



\ = ^ (m + n—l) {m -\-n — 2) — {mn — 1) = ^ (m — 1) (m — 2) 



^^{n-\){n- 2). 



This is exactly the number of arbitrary coefficients in the expres- 

 sion u^ + vyfr, so that the result follows. 



6. For a third proof, apply Jacobi's theorem to the poly- 

 nomials {x — f) w, {y — 7)) V, w, where ^, r} are the coordinates of an 

 arbitrary point. 



The points where {x - ^) u, (y — 'r])v both vanish are 

 (1) The points {Xj., y,) and here 



a {{x - ^) u, {y-v)v 



= (a-V - D (yr - V) J {^r, Vi)] 



d {x, y) 



VOL. XV. PT. v. 31 



