in relation to the theory of jjoint-groups, 481 



holds among the values of at the p- +p + 1 other intersections, 

 a^ having the value 



BioCr, yr)IJ{Xr, yr). 



This case is rather exceptional, in that u, v are not the only 

 independent (p + 1)''^'' through the points involved. We may write 



u = BV-€U, v=SW-^U 



whei-e U, V, W are p''"* and B, e, ^linear; then eW — ^V is an 

 independent (p + l)'" through the p'^ + p + 1 points, and the 

 equations satisfied by these points are* 



U V W ij = (17). 



^ ^ Hi 



In applying the converse theorem of §8 to this case it is 

 convenient to take (16) in the form 



Sttr {axr + hyr + cfP-^ = 



and integrate, say witii i-espect to b. Thus 



Sftr (a^v + byr + cy~^lyr = a function of a, c only, 



which must be homogeneous and of degree 2p—l, and can 

 therefore be represented by the sum oi' p terms of the form 



We have, then, an identity of the form 



XfSr {axr + hyr + c)'-^-' = 0, 



containing ^^ + 2p + l terms, in p of which yr = 0. The theorem 

 of §8 is therefore applicable, m, n being each =p + 1. 



* A relation of the same kind as (16) can be I'ound among the points where the 

 determinants (17) vanish, even when neither row is linear. 



