64 



ON THE EQUATION OF CURVES 



((.) When the pole is at the origin of co-ordinates we have 

 xi r= 0, yi = 0, and the equation of the polar becomes 



By + Ex + F^zo (40). 



(CO If the co-ordinates of the pole satisfy the equations, 



B^^i + Cori + E=o, D^^, + Exi + F=o (a), 



equation (39) becomes y=o, and the polar is the axis of ar. 

 Hence equations (a) determine the pole of the axis of x. 

 (>?.) If the co-ordinates of the pole satisfy the equations 



A^^i + Bx, + lD=o, Dyi + Ea:, + F=o (b), 



the polar is the axis of y, and therefore equations (b) deter- 

 mine the pole of the axis of y. 



XIII. 



Let X\ yi denote the co-ordinates of the pole of the 

 straight line 



y =z mx + h (a); 



then, since the polar of the point j;i yi is 



(A3/1 -f Ba;, +D)^ + (By 1 +Ca;i +E);r+D3^i + Ea;i+F=o,(b), 



the straight lines (a) and (b) are identical, and we have 



(A^i + Bxi -f- D)m + By, -j- Cx, + E=o (41), 



(Ay ,+ Ba;i + D) A + Dyi -|- Ea;i + F=o (42). 



(a.) When m and h are given constants, these equations 

 enable us to find the pole Xi y\ of the straight line (a). 



(/3.) When in is constant and h variable, equation (a) denotes 

 a series of lines parallel to the straight line y=7» x; and equa- 

 tion (41) shows that the pole of any of these lines lies on the 

 diameter 



(Ay+Ba;+D) m4-By+Ca;-hE=o. 

 Hence if a system of straight lines be drawn in the plane of a 

 curve of the second degree parallel to a given line, the locus of 

 their poles is the diameter which bisects chords parallel to 

 that line. 



(y.) Let equation (a) denote a system of straight lines pass- 

 ing through a given point x' y\ then 

 y' = m x' + h, 



