1887.] A. Mnkliopadhyay — Memoir on Flane Analytic Geometry. 347 

 value of the definite integral, we have 



so that, from (184), we have 



\ ^^/ an^ 



whence, finally, 



/"l \ 9. 



(187) 





Hence we have the theorem that, if we take any point on the axis of a 



parabola whose distance from the vertex is na, the sum of the squares of 



the reciprocals of all the perpendiculars dropped from this point on 



2 

 successive tangents to the parabola is equal to -r-5- I^ is obvious that 



n'^a 



these perpendiculars are the radii-vectores of a pedal of the parabola ; 

 hence, the following theorems may be enunciated. 



Theorem I. — A is the vertex and S^ the focus of a parabola whose 



latus-rectum is 4a ; points Sg, Sg, S-^ are taken on the principal 



axis such that AS^ = S^ 83= ... =a;. the sum of the squares of the 

 reciprocals of the radii-vectores of the pedal of the parabola with regard 



to Sn is -^. (188) 



Theorem II. — The sum of the squares of the reciprocals of the 

 radii-vectores of all the pedals of the parabola with regard to S^, S3 . . . Sqq is 



=M^^ ym' <-> 



Theorem III. — If we take only the odd pedals, the sum of the 

 squares of the reciprocals of all the radii-vectores is 



"Kr.+r.+ )=K:)" <-> 



Theorem IV. — If we take only the even ])edals, the sum of the 

 squares of the reciprocals of all the radii-vectores is 



=!(^^ H(i)' <»" 



§. 32. A Geometrical Locus. 



§.32. General Theorem on Conies.— If from any point P two 



tangents be drawn to the conic 



a^ + %-' (192) 



to investigate the locus of the middle point of the chord of contact when 



