332 A. Mukhopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 



7] = 



XHY2 /a;"' -2 7/«^-2' 



0.'?/. «^'^ 



/x^_y^\ 



(118) 



If, now, we eliminate a: and y between the equations (117) and (118) by 

 virtue of the relation 



m /.,v m 



\a) "^W -^' 



we shall obtain the equation of the locus sought. For this purpose, we 

 find that 



HTM-^) 



in 



= H F^,Zi -SF^ h (119) 



I abxy \ ' I I 





and 



I \m~'i- / *v / ^ \m — l 



M-B -^<-^) 



= ^' FT„-^^ — zr.-^ (120) 





(^ L. a"" 6"" -JJ 



Therefore, finally, replacing (i, rj) by (^, ?/), we find from (119) and (120) 

 the 



Theorem. — The reciprocal polar of the evolute of 



is the curve 



1 



f m in "^ m 



„ m — 1 /'^/V'm — 1 

 A;2 



/ j£? \m-i / y \m-i 



1 1 



m—l / 11 \m — 1 



= (fc.)(:p^) +(a.)(tf^) (121) 



where 7c is the radius of the circle of inversion. 



A host of interesting results may be obtained by assigning particular 

 values to on and k in (121) ; a few are noted below. 



