A CERTAIN QUADRATIC FORM 233 



Where p,-/ = P,Py are the mutual distances of the points Pi, p — OP, and 

 Pi = OPi are the radial coordiantes of the point with respect to P.-. 

 The numbers kt are the mass or ratio coordinates of P and are the content 

 coordinates when the correspondence is one-to-one. 



The equation (4) expresses the fact that p is the vector sum of ktpi. 

 The points P.- and being fixed then A-,p£ = «< are the components of 

 OP parallel to OPt respectively. Omitting any one (OPn) of OPi and 

 taking the others as axes of cartesian coordinates, any equation in the k's 

 is transformed into cartesian (a) coordinates by writing for the k's respec- 

 tively 



^, . . . ,^M_y^. (5) 



Pl Pn-l 4^ Pi 



Geometrically equation (4) is the generalization of Stewart's theorem 

 in elementary geometry. Given in a form similar to (4) by W. J. Curran 

 Sharp, Proc. Lon. Math. Soc, x-viii, p. 325, April, 1887. 



With reference to a fixed system Pi equation (4) is, if p is constant and 

 fixed, the equation of a circular locus with center and radius p in fc- 

 coordinates, the radial coordinates of the center being p,. If the k's are 

 fixed it is a circular locus with center P and radius p in radial coordinates 

 Pi. If P = then p = and (4) expresses the identical relation which 

 exists between the radial and fc-co6rdinates of the same point. It can easily 

 be shoTvm that the equation in k's 



p = Zki Pi - l^kikiPii", (6) 



is that of a circular locus whatever be the numbers p and pi, that p is the 

 power of P and pi that of P.- with respect to the circular locus. 



= Sfcipi - -Lkikjpi,^. (7) 



The Pi may be positive or negative. We say that a real circular locus 

 is orthogonal to one with imaginary radius when the former bisects the 

 boundary of the real circular locus having the same center and radius equal 

 in absolute value to that of the radius of the imaginary locus. The power 

 of a point with respect to a circular locus with imaginary radius is always 

 positive and equal to the square of the distance of the point from the center 

 plus the square of the absolute value of the radius. In case Pi is a simplicis- 

 simum, (7) is a circular locus which cuts orthogonally the circular loci hav- 

 ing centers P.- and radii 1 pj. 



