ON THE THEORY OF POINT-GROUPS. 73 



respectively, is mn — {{n)) + 1 + mn — {{m)) + ((m—n)} + l=mji— 3?i + 1. 

 This result is expressed geometrically thus : 



In order that mn points may lie on two algebraic curves of orders m, n 

 it is necessary that mn — 3n+l equations of condition shoidd subsist 

 among their co-ordinates. 



And by comparison with a previous theorem we see that if m=n the 

 number of the equations of condition is increased by one. 



A geometrical application of the above-mentioned consequence of the 

 two hypotheses is : 



If mn 2}oints are taken on a given curve of order n, n<m, there must 

 be mn — ((m)) + ((m — n)) + l=((n— 3)) equations of condition among the 

 co-ordinates of these points in order that these points may lie on a curve of 

 order m. Or in other words, 



The maximum number of j^toints ichich can be ass2tmed on a curve of 

 order n (n <m), in order that a curve of order m may pass through them, is 

 mn— ((n — 3)), which (reversing w and m) is a slightly different version 

 of Pliicker's theorem, and is established by strictly algebraical reasoning. 



Special instances of this theorem are : 



If m points are assumed on a straight line, or 2m jwints on a conic, it 

 is possible to draiv a curve of order m through them. 



If ^m points are assumed on a cubic, where m>3, one equation of con- 

 dition must hold among the co-ordinates of the points, in order that it may 

 be possible to draio a curve of order m through them. And so on. 



When m=n Jacobi obtained, as has been said, directly from equa- 

 tions (A) a system of 2((?i — 3)) equations of condition among the 2«2 

 simultaneous roots of two equations of order n. When m^n it becomes 

 a much more complicated matter to actually obtain the corresponding 

 mri—2,n + \ equations of condition. The first step Jacobi makes towards 

 this end is interesting, as it brings into consideration (although only in 

 the special case of r=«i + ?i— 3) the question of the number of arbitrary 

 constants in the equation of any curve of given order r through the points 

 of intersection of two other given curves of orders m, n, a question which 

 is of fundamental importance for the theory of point-groups. 



Given two equations, f{x, y)=0, <p{x, y)=0, of orders m, n, Jacobi 

 takes, namely, any third equation of order m + n— 3 such that it vanishes for 

 the mn simultaneous pairs of values of x, y which satisfy /=0, 0=0. Let this 

 equation be denoted by ^p^^ xy^=0, a+/3 ^ ot + m— 3, and obtain from 

 it 7nn equations by substituting in it the values of the mn pairs of simul- 

 taneous roots ; multiply these equations in order by ^ — . . . and 

 add, the result is 



^mn 



\ y^i , a;°2 y% 



R, +11, + • • •~R^;=^' 



which is the sum of all equations (A) multiplied respectively by p^g' 

 This proves that one of equations (A) is a consequence of the remainder. 

 But this is true for each and every linearly independent equation of order 

 m + TC— 3, which can be formed in such a way as to be satisfied by the mn 

 pairs of simultaneous roots ; and the number of these is equal to the num- 

 ber of arbitrary constants (homogeneous) in the equation of any one. But 

 such an equation can be formed by multiplying the left-hand side of 

 /=0 by an arbitrary function of degree (n — 3) and then adding it to the 



