OF THE SECOND DEGBEE. 



55 



and by substituting the values of m and h given by equations 

 (41) and (42) this equation becomes 



(Ay, + B;ri + D)/ + (B^, + Caj. + E) «' + D?/, + Ea: 1 +F=o, 

 which is the condition that the point x\ y\ may be on the 

 straight line 



(A2/'+Ba:' + D) 2/+ (By' + Ca;'+E) ar4-Dt/'+Ea;'+F=o. 

 Hence if a system of straight lines (a) pass through a given 

 point (x' y' )y the locus of their poles is the polar of that 

 point. 



(8.) Conversely, if any number of points lie on a straight 

 line, their polars intersect in the pole of that line. 

 Let the equation to the straight line be 



y=^mx+h (c), 



and let Xi y\ be any point on this line, so that 



yi = m Xi + h (d); 



then since the polar of Xi yi is, (xii), 



(Ayi +Bx'i +D)y + (Byi 4-Ca;i+E)x+Dyi +Exi +F=o, 

 we obtain by eliminating yi, 



{(At/ + Ba? + D) w + By+Ca; + E}a;i, 

 + (A2/ + Ba;+D) A+Dy+Ea;+F=o. 

 Now when x\ y\ is any point on the straight line (c), x\ will 

 be indeterminate, and the last equation shows that the polar 

 of any point on the straight line (c) must pass through the 

 intersection of the straight lines 



(Ay+Ba;+D)w+B2/ + Ca;+E=o^ ,. 



(Ay+B:c+D) h +Dy+Eaj+F=o; ^^'' 



which, by equations (41) and (42), is the pole of the straight 

 line (c). 



XIV. 



The forms of the principal curves represented by the gene- 

 ral equation (1) have been investigated in No. iv., and to 

 complete the discussion there given we may now consider the 

 case in which B^ = AC, and BD=AE. Tn this case we have 

 BE =: CD, and the values of x-i and y% given by equations 



