ENVELOPS. EXAMPLES. 367 



22. Find the locus of the ultimate intersections of an ellipse 



which touches a given straight line at a given point 

 at the extremity of the axis minor, the excentricity 

 varying as the axis major. Find the limits of the 

 excentricity in order that two consecutive ellipses may 

 intersect. 



23. A straight line is drawn from the focus to any point of 



a conic section, and a circle is described on it as a 

 diameter : shew that the locus of the ultimate inter- 

 sections of all such circles is a circle, except, in a 

 certain case, where it is a straight line. 



24. Shew that the locus of the ultimate intersections of all 



the chords of an ellipse which join the points of con- 

 tact of pairs of tangents at right angles to one another 

 is a confocal ellipse. 



25. Find the locus of the ultimate intersections of the straight 



lines x cos 30 + y sin 30 = a (cos 20)^, where 6 is the 

 variable parameter. 



Result, (a; 2 + 2/ 2 ) 2 = a 2 (x* - y*}. 



26. Find the envelop of the circles described on the radii of 



an ellipse, drawn from the centre, as diameters. 



Result. (a 2 + 2/ 2 ) 2 = a 



n 



27. On any radius vector of the curve r = c sec n - as diameter 



is described a circle : shew that the envelop of all such 



a 



circles is the curve r = c sec n ~* . 



n 1 



28. Find the locus of the ultimate intersections of a family 



of parabolas of which the pole of a given equiangular 

 spiral is the focus, and its tangents directrices. 



Result. A. similar equiangular spiral. 



