in relation to the theory of point-groups. 479 



If then w (ocr, yr) = 0, {r = l, 2 ... mn), we have from (8) 



w = Pu + Sv 



and P, 8 can no longer be fractional, since w is not fractional and 

 the denominators of P, 8 have no common factor. 



Hence any polynomial w of degree ^m-f?^ — 1, which 

 vanishes at (xr, y,) (^ = 1, 2 ... mn), must be of the form u^ + v-^ 

 where <^, i/r are polynomials of degrees n — l,'m — \ respectively. 



7. If lu is of degree m + w — 2 only, the terms of degree 

 m + n — 1 in ucf) + v\}r must cancel, which can only happen if the 

 terms of degrees w — l,m — 1 in (j>, ylr vanish identically, since the 

 curves u, v have no intersection at infinity. By applying this 

 argument repeatedly we find that the degrees of <^, yjr are 

 r — m, 1 — n where r is that of w. 



To extend the theorem to higher values of r than those for 

 which it has been proved, it is only necessary to note that when 

 r > m + w — 2, homogeneous polynomials </>!, -^i of degrees 



r — m, r — n 



can be found such that the highest terms in wc^j + v^^^ coincide 

 with those in w, so that by subtraction the degree of w is 

 reduced; when r = m + n — 2, the degree can be reduced by 

 subtracting Ui^^ + v-^^ +cJ* , where the polynomials <^i, i/^i and 

 the constant c are suitably chosen. By applying Jacobi's theorem 

 to the reduced expression w — u(f)i — v-yjri — cJ, whose degree is 

 m + n — S, we find — mnc = 0, so that c must be zero, if w vanishes 

 at all the intersections. 



8, Converse of Jacobi's Theorem. Suppose now that mn 

 points (xr, yr) {r=l, 2 ... mn) are such that, for any polynomial 

 w of degree m + n — S, 



X arW(Xr, 2/r) = (9), 



the coefficients a^ being independent of those in w, and let us 

 investigate whether these mn points are necessarily the complete 

 intersection of two curves of degrees m, n. 



We may also take the conditions involved in (9) in a form in 

 which they have been discussed by Serret, Sylvester, Clifford and 

 others, namely 



S ar(aXr + hyr + cy''+''-' = (10), 



for all values of a, h, c. 



Divide the mn points into two groups, A and B, containing 

 respectively ^m {m + 3) and |-m {2n — m - 3) points. Let \ be 

 the number of independent m"' vanishing at A and I tlie 



* 8ee for instance Camh. Phil. Froc. vol. 14, p. 389. This step is not necessary 

 if the proof of § 6 is used. 



