CONIC SECTIONS, ANALYTICAL. 



VII. Envelopes. 



The equation of the tangent to a parabola in the form 



a 

 m* 



gives as the condition of equal roots in m, y* = 4ax, and the 

 equation of the tangent to an ellipse 



c y f 



- cos a + J- sin a = 1, 

 a 6 



written in the form 



s as the condition of equal roots in z, 



So, in general, if the equation of a straight line, or curve, involve 

 a parameter in the second degree, it follows, that through any 

 proposed point can in general be drawn two straight lines, or 

 S of the series represented by the equation. These two 

 curves will be the tangents (rectilinear or curvilinear) from the 

 proposed point to the envelope of the system. In order that 

 may be c' the point from which they are drawn must be 



a point on the envelope. 



Thus, "To find the envelope of a system of conies having 

 a given focus, length and direction of major axis." 



The equation of any such conic may be taken 



where is the parameter. The envelope is then the curve 



