A CERTAIN QUADRATIC FORM 



235 



(11) may have any signs. If we call r the mean radius of a curve as to its 

 area so determined that ttH is its area, then (11) can be written 



r^ = Sxri" — Ilxyrj2-, 



(12) 



in which the r's may be real or imaginary. For constant r's (12) is the 

 equation of a conic ■nath respect to any fixed position of the triangle 

 P1P2P3, it is of the eUipse, h3rperbola, parabola class according as 



5 = s(s — rn) (s - raa) (s - nO, 



where 2s = ?-i2 + ros + nn is positive, negative or zero. In particular it 

 is a circle when ?■,■,■ are respectively proportional to the sides of the triangle 

 P1P2P3. 



The conic is two straight lines when 



D= 



(13) 



is zero. The straight lines are imaginary, intersect or are parallel accord- 

 ing as S is positive, negative or zero. 



In the case of a closed conic Vij are the sides of a triangle, say UVW, 

 and with respect to this triangle (12) is the equation of a circle, and for 

 different values of r a series of concentric circles. The constants rr, ri^, ri 

 are the powers of TJ, Y, W with respect to the circle 



= "ZxTi^ — lixy ris~, 



(13) 



and r' is the power of x, y, z with respect to (13). When D = (13) 

 shrinks to its center becoming the point circle, or two imaginary straight 

 lines intersecting in the real point P whose radial coordinates are ri, r^, n 

 and areal are x, y, z with respect to UVW. When the points U, V, W are 

 colinear (12) is a parabola, and in particular two parallel straight lines when 

 ri, ?"2, '"3 are the distances of a point from U, V, W. 



If the moving triangle P1P2P3 in the xy-ip\a,ne is a rigid triangle ABC 

 the equation (11) becomes* 



(P) = 2x(A) - Tvlxyc- 



(14) 



* In case the triangle returns to its initial position without complete revolution 

 the last term Sxy (P12) on the right is zero. 



