1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 341 



Let CD be the polar of G with respect to the conic, so that CD is parallel 

 to the axis-major and has for its equation 



_ fcs 



^^ "" c2 sin (/>* 



Transfer the origin to P, and take the new axes parallel to the old ; 



then the ellipse is 



(x-{-a cos <p)^ (y-\-if sin <^)2_ , 

 — -|- — — j_ 



x^ 1/2 2 cos 2 sin 



r> + P+-T-^+-r^^=" (1^^) 



The line CD is 



y-\-h sin</>= -■ „ . . , 



or 2/ = ^ (162) 



where A = - „— ^— -- (a^ sin^ (^ + fc^ cos^ <^) (163) 



c2 sm <^ 



Now, PD, PC are two lines through the new origin, and through the 



intersection of the conic with the line ; their equation, therefore, must be 



x^ ^2 2 cos <l> 2 sin <^ „ ^ .. ^ ,. 



^>+P + ^:^^^+-xy-^^=^ (16*) 



These will be at right angles, if 



JL ^ 2^in^ 



Substituting for A, from (163) and simplifying, we have 



sin»^.(l-i)(l-|) (165) 



which determines the value of <p, and, therefore, of P ; it is remarkable 

 that the result is dependent simply on the eccentricity. 



III. A very interesting point arises, if we seek the envelope of the 

 sides of any triangle PSS' having its vertex P at any point on the ellipse* 

 and its base-ends any two points S, S' on the axis-major, equidistant 

 from the centre, so that OS = OS' == k. Then, from (150), the equation 

 of PS is 



b sin <l> bh sin 



a cos 'p — h cos ^ — Ic 

 which may be written 



(bx — ahb) sin 'p — ay cos <l> = — ahj, 

 and the envelope of this for different values of <p is 



{bx-akby-^ah/^a^Jch/^ (166) 



which is equivalent to 



b^ (a-ak)^=:a^ (k^-l) y^ (167) 



or b (x— ak) = ± a s/ k^ — 1 y \ 



