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



nates of A, A', S, S', P are (a, o), ( — a, o), (ae, o), ( — ae, o), (a cos ^, 

 6 sin </>), respectively. The equations to PA, PS, PS', PA' are easily 

 found, viz. J 



. X — a cos <p _a cos ^ — a, 



PA is , — : T = 7 : — "T J 



y — o sm <p b sm <p 



h sin </) 6sin</> ' 

 or ^/ = T — r^ T — T (149) 



a cos f — l cos 9 — 1 



_ -, . jc — tt cos <^ a cos <p — ae 



PS is — r--^— :! = —J — -—r- ' 

 y — sm 9 o sm <p 



6 sin 9 5e sin </> 

 or y = X (loO) 



a cos f — e cos <f> — ^ 



■001/ • ^ — a cos ^ a cos ^ -f «.e 



Pb is r— : — ; = — 7 — • — 7 — J 



y — sm 6 sm <^ 



6 sin 9 6e sin <^ 



or 2/ = 7-r-^'+ TT- (1^1) 



a cos 9 + e cos 9+ e 



^ , , . a? — a cos </> a cos </> + a 



PA' is r— : — 7=-^ — ^:r~' 



y — sm <p sin f 



b sin 9 & sin </> 



^ a cos 9 + 1 cos 9 + 1 

 Let jp, q be the intercepts made by PA, PA', and r, s those made by 

 PS, PS', on the minor axis. Then we have 



b sin </> _ 6 sin 9 



•^ ~ 1 - cos"^' ^ ~ l+cos~^' 

 6e sin 9 6e sin </> 



""e — cos<^' ~~e-hcos^' 

 so that we get 



•^ * sm 9 ■* -^ ^ ^ 6 sm 9 



2be"sin9 6V sin»9 11 2 



e* — cos^ <^' e^— cos^ <l>^ r s b sin 9* 



This shews that the sum of the reciprocals of the intercepts made by 

 PA, PA' on the minor axis is equal to the sum of the reciprocals of the 

 intercepts made by PS, PS' on the same axis ; it also follows that, since 

 pq = b^, the rectangle under the intercepts made by PA, PA' is always 

 constant and equal to the square of the semi- axis- minor. Again, p, q are 

 the roots of the quadratic 



i2'-2& cosec^. ;2 + 6'' = (153) 



Similarly, r, s are the roots of the quadratic 



^''-26 A'cosecf « + 6»A' = (154) 



where \^ satisfies the equation 



_ e" sin" 9 

 e^ — cos" ^* 



