1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 837 



tlie coordinates of the corresponding point on the reciprocal polar of the 

 evolute, may be similarly expressed. For, remembering that 



du 



dx dy 



du dx* 



dy 

 the formulae in (147) and (148) may be written 



dx 



^ = F. 



^ = ^^ 



ytx^' 



dy 



dx 



dy . 



= kK 



= kK 



dy dx 

 ^ # d<l> 



dy 



d^ 

 dy dx^ 



so that, if the coordinates of any point on the given curve be given by 



we see at once that the coordinates of the corresponding point on the 

 reciprocal polar of the evolute are given by the system 



/i m 



i^k' 



AW 



It is clear that the coordinates of any point on the n^^ " reciprocal polar 

 of evolute " may be obtained from this system ; and the coordinates of 

 points on the curves given above may also be expressed by means of a 

 single variable parameter. 



§§. 28 — 29. Theorems on Central Conies. 



§. 28. Properties of the 



Ellipse. — In this section we 

 shall investigate the truth of 

 some theorems on the ellipse. 



I. Let <^ be the eccentric 

 angle at any point P on the 

 ellipse 



^a ^ T,a ~ ■*•' 



or 



so that, if A, A' are the vertices 

 and S, S' the foci, the coordi- 

 43 



