1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 331 

 §. 27. Reciprocal of Evolute of Conic.*— We now purpose to 



investigate the reciprocal polars of evolutes of couics; but as all central 

 conies are included in the equation 



©"-©■=■• <-> 



we will discuss the problem with regard to this general case. Since the 

 reciprocal is the inverse of the pedal, and as the pedal of the evolute is 

 the locus of the intersection of the normal and the line drawn at right 

 angles to it through the origin, it is clear that the reciprocal polar of 

 th.e evolute is the inverse of the locus of the point of intersection of 

 the normal at any point of the curve, and the right line dropped per- 

 pendicular to it from the origin. Now, the normal at any point (^, y) 

 of the curve in (112) is 



, w — 1 m — 1 



where X, Y being the current coordinates, the equation may be written 

 ^ Y-'L X = ^-2/(f -^) (113; 



The straight line through the origin at right angles to this, is 



m-1 /r^^-l 

 y Y + f X=^0 (114) 



At the common point of intersection of the two lines given by (113) 

 and (114), we have 



If a, 7j) be the inverse of the point whose coordinates are given by 

 (115) and (116), and k^ the constant of inversion, we have 



i=Jf^- ^'liTi (117) 



xy. 



/^^^m-2 y,n~2 \ 



* The theoreins established in this section were discovered by me about three 

 years ago, and were, on the 29th August, 1885, communicated to Mr. W. J. C. Miller, 

 Mathematical Editor of the Educational Times, with a view to their publication in 

 that journal. Thej have since been published as questions 8571, 8707, 8773, 890:5, 

 9049, 9074, 9148, 9162, 9163, 9204 ; but, while some of these questions have appeared 

 under my name, the others have been, for reasons best known to Mr. Miller himself, 

 ascribed to different gentlemen who had, perhaps, just as much to do with the 

 theorems with which they have been credited, as the proverbial man in the moon. 



