368 ENVELOPS. EXAMPLES. 



29. Perpendiculars are drawn from the pole of an equi- 



angular spiral on the tangents to the curve : find the 

 envelop of the circles described on these perpendiculars 

 as diameters. 



Result. A similar equiangular spiral. 



30. From every point of a parabola as centre a circle is 



described with a radius exceeding the focal distance 

 of the point by a constant quantity : find the envelop 

 of the circles. 



Result, (x + c + a) [y* + (x a) 2 c 2 } = ; where c is 

 the constant quantity. 



31. Find the envelop of the straight lines obtained by vary- 



ing 6 in the equation ax sec d ly cosec 6 = a 2 J*. 



Result (ax] + (ly)* = (a 2 - J 2 )l 



32. From a fixed point A in the circumference of a circle 



any chord AP is drawn and bisected at H, and on 

 PH as diameter a circle is described : find the locus 

 of the ultimate intersections of the system of circles 

 described according to this law. 



Result, a 2 (x* + y*} = (2x* + 2y*- 

 where a? 4- y* = 2ax is the equation to the given circle. 



