OP THE SECOND DEGKEE. 53 



tan.^^ ;V/(B^-^Q (37). 



A + C — 2 B cos 7 ^ ' 



From this equation we readily deduce 

 sin 6 2 / (B2 — AC) 



sin y l/ (A + C — 2 B cos yf + 4 (B^ — AC) sin V 

 and by substituting this in equation (a) we get 



c^=jp^^/((A + C— 2Bcosy)2+4(B2— AC)sinV}(38). 



The constant c, determined by this equation, is sometimes 

 called the power of the hyperbola. 



XII. 



Any point ari 3^1 being given in the plane of the curve (1), 

 the straight line whose equation is 



( A^, + Ba;, + D)3/ + (By , + Car, + E)ar + Dy 1 + EoJi 4- F = . . . (39) 

 is called the polar of the point ajj y\ in relation to the curve 

 (1), and the point Xx y\ is called the pole of the straight line 

 (39). From these definitions the following theorems are 

 immediately obvious. 



(a.) When the pole is on the curve (1), the polar passes 

 through the pole and touches the curve at that point, (x). 



(^.) When the pole is without the curve, the polar is the 

 chord of contact of the two tangents drawn from the pole to 

 the curve (x, /3). 



(y.) When the pole is within the curve, the polar is the 

 locus of the intersection of two tangents applied to the curve 

 at the extremities of any chord passing through the pole. 

 This is also true when the pole is on the curve (1) or outside 



of it (x. y). 



(S.) When the pole is at the centre of the curve (1), 

 Kyx + Ba;, + D = o, B^/, + Ca^i + E=o, (ii, y), 

 and the equation of the polar becomes, (iv), 



o.y + o. x + F"=o; 

 hence when the pole is at the centre of the curve the polar is 

 at infinity. 



