324 A. Mukhopadliyay — Memoir on Plane Analytic Geometry. [No. 3, 



reciprocals of the rectangles under the segments of each chord is constant, 

 the variation of this constant furnishing the different members of the 

 family of similar conies. 

 Let 



ax^-\-'2.1ixy-irly'^ + 'lgx-\-2fy-\-c = (95) 



be the primitive conic, and {x\ y) the point through which the chords 

 are drawn at right angles to each other and whose locus we seek. 

 Transferring the origin to this point, the conic becomes 



ax'^-\-21ixy-\hy'^-\-2g'x-^2fy^c' = (96) 



where c' is the point-function. The polar form of this equation is 

 (acos2^-+2/icos6'sin^ + &sin2(9)p3 + 2(^'cos(94-/sin^)p-|-c' = ... (97) 

 Hence, if pj^, pg ^® ^^^ segments of the chord drawn through the new 

 origin, inclined at an angle 6 to the axis of x, and pg, p^ the segments of 

 the chord at right angles, we have, from (97), 

 _ c 



^1 ^2 ~ a cos2 ^+2/i cos e sin 6-^-1 sin^ (9' 



^^ ^* ~ a sinS^ - 2/i sin ^ cos 61 + & co^W 



so that 



1 1 _aVb 



P\ P2 PZ P4 ^ 



which shews that the sum of the reciprocals of the rectangles is in- 

 dependent of the direction of the chord, and for any given value of this 



sum, say — , the locus of {x\ y') is given by 



a^-h _ 1 

 c' ^A2' 



which may be written 



aa;2+2A^?/ + %H 2^;r+2/^-f c = 7^?^ (a + &) (98) 



and this, of course, represents a conic concentric with the primitive one 

 given by (95), and similar and similarly situated; and we get a 

 family of similar conies by assigning all possible values to h. It 

 is interesting to remark that the property established here is general 

 in a twofold sense, viz., if the sum of the reciprocals of the rectangles 

 under the segments is to be constant, the point may be any point on the 

 conic given by (98), and the chords may be inclined at any angle to the 

 axis of X, provided they include a right angle. The same results, of 

 course, could have been obtained by applying the process to each of the 

 conies separately, viz., if we have the central conio 



