424 



blem, 'given an alteration in the curve and assignant equations, 

 to find that in the concrete equation,' admits of a similar solution. 

 Given 



as the equations to the locus, and also 



C^p, q, r, *) = 0, C a (p, q, r, *) = 0, A = f '(p, q, r, s) + V(p, q, r, = )0 

 as the new curve and assignant equations: put = 0', ^=^', and 

 determine x = (p, q, r, *), y=r)(p, q, r, s), and then use 



as the new concrete equation. The result is independent of the 

 form of the curve equations. The geometrical significance of these 

 transformations is that there are no ' families ' of plane curves. 



' Clinant radical loci,' or curves which furnish a sensible geome- 

 trical picture of the relations of the corresponding clinants which 

 satisfy any abstract equation, may be obtained thus. Let the given 

 equation be C (X, Y)=0. This may be regarded as two reduced 

 curve equations, 



C i (P> Q> r y *) = 0, C 2 (p, q,r,s) = 0. 



The values of X are assumed by drawing radii vctorees to points in 

 any curve, and the corresponding values of Y have to be pictured. 

 We must therefore have some equation A=0, which in combination 

 with the other two will give the curve by which X is thus determined. 

 Eliminating the scalars, p, q, r, s, two and two, between these three 

 equations, we obtain the following six, of which the first and last, 

 and one of the intermediate ones, are in general only required : 



We now construct the curves containing the points X, S, Y, as the 

 loci of the equations 



OX=OP+PX = 

 OS=OP+PS = 

 OY = OR+RY = OR+PS = (r+ i . *) . OI,/ 6 (r, *)=0. 

 Set off OP at pleasure on OI (produced both ways), and draw the 

 ordinate PXS, cutting the two first curves in X and S. Through S 

 draw SY parallel to OI, cutting the third locus in Y : then if 

 OX=X . OI, and OY=Y . OI, X and Y are corresponding solutions 



