480 Mr Dixon, Jacohi's double-residue theorem 



number among these that vanish at B also. Let k be the number 

 of independent {n — 8)-ics vanishing at B and k the number 

 among these that vanish at A also. 



If 0, i/r are of the degrees m, n — 3, we have 



S a,, (ji (w,, y,) yfr (ccr,yr) = (11), 



and by taking (f) to vanish at A we have \ — I homogeneous 

 linear relations among the values of ^ff at B. It is here supposed 

 that none of the coefficients Ui, Wa ... vanish. Hence 



K>^(n-1) (n - 2) - lm(2n -m-S) + \-l ...(12), 



and similarly by taking yjr in (11) to vanish at B we have 



X ^ I (m + 1) {m + 2) - hn {m + S) + k- k (13). 



By addition 



k + l^^{m-7i + l){m-n + 2)+ 1 (14). 



Similarly 



i+j>^in- m + l){n - ??i + 2) + 1 (15), 



if i, j are the numbers of polynomials of degrees ni — S, n 

 respectively which vanish at all the points (*',., Vr)- The desired 

 conclusion can often be deduced from (14) and (15). For instance, 

 if m = n, so that i = k, j = I, we have 1^2 if k = 0; that is, the 7n- 

 points are common to two ni^'^^ unless they lie on an (m — 3)"^, 

 and similarly in other cases. The relations (14), (15) moreover, 

 as equalities, are those satisfied in general when the mn points 

 are the intersections of an m^'^ and an n^°. Still, tbis is not the 

 only possible consequence of the condition (9), as the following 

 cases shew. 



I. Take m = 8,n = 7, so that there are 56 points, and any 

 curve of degree 12 through 55 of them contains all. Take 56 of 

 the intersections of two curves of degrees 5, 12, Then any 12"^ 

 through 54 of these will contain all, and therefore they will satisfy 

 two conditions such as (9) and yet will not in general be the 

 complete intersection of a septimic and an octavic. 



II. Take to = w = 9, so that there are 81 points. Choose 

 these among the 90 intersections of two curves of degrees 15, 6. 

 Then a 15'" through 80 of them will generally pass through the 

 other one, and a condition of the form (9) is satisfied, but the 

 points are not the complete intersection of two nonics. 



Thus the converse of Jacohis theorem appears to consist in 

 the statement that (14) (15) follow from (9). 



9. If u, V are of the same degree, p + 1, and p of their inter- 

 sections are collinear we may take w to be (f)S where S is the line 

 containing p intersections and ^ is an arbitrary (2p — 2)'". Then 

 a relation 



Xar(t){Xr, yr) = (16) 



