422 



the intersection will evidently be determined by putting pp', 

 q = q', r=.r\ s=s f , which will give six simultaneous equations between 

 four variables, p, q, r, s or^/, q', r', s 1 . This gives two equations of 

 condition. If, however, no assignant equations were given, we might 

 determine the values of p, q, r, s from the four reduced curve equa- 

 tions, and then assume any assignant equations compatible with these 

 solutions. This is more readily done by determining X and Y from 

 the two unreduced curve equations C(X, Y)=0, C'(X, Y)=0. 

 The process then corresponds to that for the simplest scalar case of 

 intersection. If the values of X and Y prove to be scalar, then the 

 assignant equations are # = and $=0, and we have an ordinary 

 scalar case of intersection. But if this is not the case, and we find 



we may take 



(0-*i) .(?-6)=0, (s-^)... 0-O = 0, 



among others, as assignant equations and determine the corre- 

 sponding loci. These loci will be found to intersect in all the (per- 

 fectly real) points determined by the values of X and Y, but not 

 necessarily in these only. Such points will of course not belong to 

 the curves derived from putting # = 0, s=0, and hence cannot in 

 any sense be called points of intersection of these curves, although 

 they have hitherto been termed ' imaginary points of intersection.' 



The discovery of equations to loci described according to some 

 geometrical law, furnishes a convenient illustration of this clinant 

 theory. From any point O, draw radii vectores OU, OR to any 

 curves, and make RP=A . OU, where h is scalar. Put 



OP=O+ i . j) . OI, OR=(r+i . *) . OI, OU=(w+i . v) . OI. 



Then the condition OP=OR + RP gives p=r+ hu, q=s+hv, which 

 correspond to the two reduced curve equations. We now require 

 three more equations in order to eliminate four of the six scalars 

 P> q> r > s Uy v > an d find a relation between the two remaining scalars, 

 sufficient with one of the above concrete equations to determine the 

 locus of one of the points P, R, U. Two of these three equations 

 will amount to assigning the locus of two of these points, and the 

 third equation will amount to assigning some relation between the 

 angular motions of OU and OR, without which the loci can clearly 



