OF THE SECOND DEGREE. 



37 



point of the chord Pi Pg. Now, when m is constant, or the 

 chord P1P2 is parallel to a given straight line y=m x, it is 

 evident, from equations (c), that D'm4-E'=o is the condition 

 that the middle point Xi yi of the chord Pi Pg may be on the 

 straight line 



{Ai/+Ba;+'D)m+(By+Cx+E)=o (6). 



Hence we see that the line (6) is the locus of the points of 

 bisection of all chords of the curve (1), which are parallel to 

 the straight line y = m x. 



The straight line (6) which bisects chords parallel to the 

 line^=mj7, is called a diameter of the curve (1), and any 

 chord (2) which is bisected by the diameter (6), is called an 

 ordinate to that diameter. 



(a.) When the ordinates are parallel to the axis of a? we have 

 w=o, and equation (6) becomes 



B^+Ca;+E=o (7), 



which is, therefore, the equation to the diameter that bisects 

 chords parallel to the axis of x. 



(/3.) When m= 00, equation (6) gives 



Ay+Bx+'D=o (8); 



this is, therefore, the equation of the diameter which bisects 

 chords parallel to the axis of y. 



(y.) It is evident from the form of equation (6) that every 

 diameter of the curve (1) passes through the point of intersec- 

 tion of the straight lines (7) and (8.) On account of this re- 

 markable property the intersection of these lines is called the 

 centre of the curve (1.) Let xg and ^3 denote the co-ordinates 

 of the centre, then since the point arg yg is on each of the 

 straight lines (7) and (8), we shall have 



Ayj+Ba?24-D=o, Byg+Carj+Erro, 



from which we obtain by elimination 



AE— BD CD— BE ,^ , 



^="B^=:AC ' ^^="B^-=AC ^^-^ 



These equations indicate a very simple method of finding the 



