1904.] BROOKS— ORTHIC CURVES. 327 



where X is a real parameter. The locus of centres is 



The elimination of A from this and its conjugate gives 



the equation of a right line. 



IX. T/ie Hypocycloid Enveloped by the Asymptotes. 



Let us now seek the curve enveloped by the asymptotes of tlie 

 curves of a general pencil. The equation of an asymptote of the 

 curve given by t■^ is 



^ - K^i + A^'i) + VA ^ '^ — i (^2n-i^-' + a',,_,) !> = o, 

 or 



^^—i (^1 + A^'i) + °i^A { ^v — {a, + t,a\) )-=^o. 



For convenience, transform to the centre of the pencil, ^a, as a 

 new origin. The equation becomes 



.V— ^ ^Vi + °i/ A(^' — i«\A) = o. 

 Putting r° = /, we get 



and finally, 



X ^^V -h TJt n^^'l"'~° = Oj 



.rr-^ — i^\r»-^ + x — ^a\r-^ = o. 



Now the map equation of the curve enveloped by this line as r 

 varies is 



^!X = ?m\r'-'' 4- (i — ;/) ^\r°. 



Now this equation represents a curve of double circular motion. 

 We know that 



a\ = a\t.>-\ 

 and using it we get 



7LV = ;ia\t,-'r'-'' j^ (i — n) «V°. 



Now if we make t.^ real, and then regard the centre circle as the unit 



