SUPPLEMENTARY 0UEVE8. 



196 



curve (5) is of the eighth degree, the origin being a multiple 

 point of the sixth order. 



(a) It is evident from the forms of the equations (4) and 

 (5), that the curve whose equation is 



A^r + B, = o (6) 



bisects every chord of either curve which passes through 

 the origin A. Plence the locus of the points of bisection of 

 all chords of the curves (4) and (5) which pass through the 

 common multiple point A, is a curve of the fourth degree 

 having A. for a triple point. 



(6) By clearing the equations (4) and (5) of radicals, we 

 obtain 



(A.r + B.r = B/--A,C. (7), 



(A.r+B.r= A.C.-B,» (8); 



from which it is evident (III) that each of the supplementary 

 curves touches the system of straight lines represented by the 

 equation 



B;— A,C. = o (9) 



at the points in which these lines are cut by the curve (6). 

 Hence, also, the supplementary curves touch each other at 

 the points in question. 



(c) If a straight line be drawn from the origin A to any 

 point P on either of the supplementary curves, cutting the 

 curve (6) in the points Q^, Q,,, Q-^, Q^; and if PR^, PR,, 

 P R,, &c. be the perpendiculars drawn from P to the system 

 of straight lines (9), we should find as in No. XI. that 



PR PR PR istoi ^^^'^^^'^^^-'^^^ X 



^^'•^^^^ ^^^""^^ VAQ..AQ,.Aa3.Aay 



in a fixed ratio ; and therefore the two supplementary curves 

 (4) and (5) may be considered as branches of a geometrical 

 locus, all the points of which are determined by a unique 

 geometrical condition. 



