in relation to the theory of point-groups. 473 



have a common factor*; which is contrary to the supposition that 

 the curves m = 0, v = meet in mn isolated points. 

 There must then be a relation of the form 



■mn 



S Ar w{Xr,yr) = (2), 



r = 1 



satisfied by an arbitrary polynomial w of degree m + n — 3. 



The coefficients A can be found by constructing certain 

 particular polynomials by Kronecker's method. 



* There is a slight difference in the argument at this point when the number of 

 variables is greater than two. Suppose for instance there to be three variables 

 xi, X2, X3 and ui, u^, Uz to be polynomials of degrees mx, m2, m^ and lo to be of 

 degree mj + wi2 + WI3 - 4, so that Jacobi's theorem becomes 



/ 0{Xi, X2, Xz) 



It is possible to have polynomials 4,-^, (p2, ^3 of degrees ni2 + m3-4, ms + m^-i, 

 mi + m.2-i, such that 



i«X^l + U2<t>2 + U3<p3 = 0. 



Using a bar to distinguish the terms of highest degree we have 



Wj^l + U2<p2 + «303 = 0, 



a homogeneous relation. It is supposed that the surfaces ui, 112, u^ have no 

 common point at infinity, and hence when W2 = and 2*3 = 0, mi cannot vanish, so 

 that ^1 = 0. Thus by Nother's theorem (§ 7 below) 



<P\ = W2r/'3 - U31P2, 



where \p2 . fs are of degrees ni2 - 4, ma - 4 and homogeneous. It follows that 



W2 (^2 + "1^3) + M3 (^3 - u-^i) = 0, 

 and that ^2 = '/'1W3 - ^■^i , 



03 = ^2"l-'/'lW2. 



where \j/i is homogeneous of degree nii- 4. 

 Hence in the identity 



"101 + "202 + "303 = 0, 



01 ) 02 » 03 may be replaced by the polynomials of lower degrees, 



01 - W2'/'3 + «3^2 . 02 - W3\^l + "l 03 , 03 " "l 02 + "201 , 



and the degrees of these may be lowered similarly until we arrive at the result 



01 = "203 -"302 J 02 = "301 -"103. 03 = "l 02 " "201 , 



where 0i, 02, 03 are of the degrees mj - 4, w(2 - 4, m^ - 4. 



Thus if 01, 02, 03 are arbitrary polynomials of their degrees the effective number 

 of arbitrary coefficients in 



"101 + "202 + "303 



is the number of coefficients in <pi, 02, 03 diminished by the number in 0i, 02, 03, 

 that is, 



^[(m2 + 7ft3-l) (m2 + m3-2)(m2 + ni3-3) + (?n3 + mi-l)(»n3 + mi-2)(m3 + mi-3) 

 + {mi + 7«2 - 1) (toi + m2 - 2) {mi + m2 - 3) - (mi - 1) (mi - 2) (mi - 3) 

 - (m2 - 1) (m2 - 2) (m2 - 3) - (mg - 1) (mg - 2) (mg - 3)], 

 or |(mi + m2 + m3-l) (mi + ??i2 + m3 - 2) (mi + m2 + wi3-3) -mim2m3 + l. 



