of Curves and Curved Sutfaces of the Second Order. 543 



stance of the major directrix from the centre is to the square 

 of the abscissa a of the point of contact, . juj « «i9qoIy/iii» ii ^i 



or" O Q . O Q' : OP^ : : ^ :i^*^ '^"^"^ "^^' "^ ^'^*^ 



" '5C!. Let the lines O Q, O Q' meet the same minor directrix 

 in the points Q Q", then in the triangle O Q Q" we shall have 

 the base Q Q' to the sum of the sides O Q + O Q", as the 

 eccentricity of the conic section is to unity, or 



QQ" • . u u . QQ" f"^ -^^ 



XI. On this tangent let perpendiculars be let fall from the 

 mhurrfoci C C, the ratio of these perpendiculars is equal to 

 that of the distances of Q and Q' from the centre. Hence 

 the ratio of those perpendiculars is also the same as that of the 

 segments P Q, PQ'; thus the perpendiculars from the minor 

 foci on a tangent, the distances of the points Q Q' (where this 

 tangent meets the minor directrices) to the centre, and the 

 segments of this tangent between the point of contact and the 

 minor directrices are all in the same I'atio. 



XII. The product of the focal perpendiculars on a tangent 

 to the curve, is to the square of the central perpendicular on 

 the same tangent as the square of the semi-diameter passing 

 through the point of contact is to the square oi^ the semirmajoi" 

 axe, ^ ■• 



or -pa- - -^• 



Let X be the angle which these perpendiculars make with the 

 minor axe, and let p p' be the distances of the points Q Q' to 

 the centre, 



then — r = ^^ sin^ a. 



?? 



XIII. From any point G in one of the minor directrices 

 let tangents be drawn to the curve, and let the cord of contact 

 meet ll)e same directrix in the point H, the line G H subtends 

 a right angle at the centre O. 



XIV. Or more generally, if from any point G in the plane 

 of the curve, two tangents are drawn to it meeting one of the 

 minor directrices in the points m and 7/, the cord of con- 

 tact meeting the same directrix in the point H, the line O H 



'i^^bisects the supplement of the angle /« O w ; and if from C, the 

 corresponding minor focus, we draw a line to the point G 

 meeting the directrix in the point H', the line OH' bisects 

 the angle mOn\ hence the lines O H, O H' are always at 

 right angles to each other. 



