384 Prof. Sylvester on the Analogues in Space 



points at infinity for double points, and being doubly touched* 

 by each line joining either of them with any one of the four fixed 

 points; and if we are at liberty to assume (which, however, 

 requires further investigation f) that the curve containing the 

 foci of U -f XV must be identical with one of the group, then this 

 curve of foci will be defined by the equation 



/ . \/A + m \/B + n\/C +p \/D = ; 

 whence, calling a, b, c, d the four fixed points, and F, G, H the 

 points of intersection of the opposite sides of the quadrangle 

 a, b, c, d, the signs of the square roots below written must be 

 capable of being so assumed that the determinant 



(«P)*j (bFji 



(aG)*; (4G)* 



(oH)*; (6H)i 



1 ; 1 



shall be equal to zero, 



(cF)*; {d¥)i 



(cG)i; (dG)i 



(cH)*j {dRf 



1 \ 1 



constituting a remarkable theorem concerning seven points (four 

 quite arbitrary) in a plane. If (as seems probable) the case 

 supposed is what actually obtains, a geometrical rule must 

 exist for determining the proper combination of signs to be em- 

 ployed in the above determinant ; and then /; m; n; p will be 

 proportional to the first minors of the three first lines of the 

 matrix above written J. 



The above theory, very hastily sketched out under the pressure 

 of other occupations, will serve at all events to manifest in how 

 very imperfect and incohate a form the theory of foci at present 

 exists, and may serve to raise the question whether there is not 

 a family of curves distinguishable by the possession of syzygetic 

 foci, and which may be termed syzygetic or norm curves (in- 

 cluding as elementary members of the group conies, Cartesian 



* In general, if a curve is defined by a homogeneous linear relation be- 

 tween its distances from r, or a non-homogeneous linear relation between 

 its distances from r—\ points, it will easily be seen that the line joining 

 each such point with either circular point at infinity will be a tangent touch- 

 ing the curve in 2 r ~ 2 points. 



t To the group of syzygetic curves there are only eight parameters, and 

 these eight tangents (all double) at the four foci are common to each mem- 

 ber of the group and to the locus of the foci of U+^V. To complete the 

 proof of the supposed identity of the latter with one of the former, it is 

 necessary to show that the number of curves having the eight tangents in 

 question is precisely equal to the number of systems of solutions of the 

 equations l+m+n+p = 0, 



L=0, M = 0. 



% Calling the determinant D, the equation D = serves to fix the allow- 

 able combinations of the doubtful signs, just as in Cardan's rule a certain 

 equation, which the product of the two associated cube-roots in the solu- 

 tion is bound to satisfy, fixes their allowable combinations of values. 



