1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 327 

 and this may be written 



r |cos A + cos (^- k-2e\] = E cos ^ + F sin 6, ... (105) 

 "where 



E = 2sin A |«;sin ^-^-AW?/sin^ j (106) 



F = 2 sin A j 2/ cos ^-a? cos (-^- A\ j (107) 



Eliminating r between (102) and (105), we have 



2c I cos A+cos (^ - A -2^y| 



= PE. 2 cos2 ^ + QF. 2 sin2 ^-|-(QE4-PF). 2 sin 6 gob 9. 

 Assuming, therefore, 2^ = <^, this may be written 



2c cos A -f 2c cos (- — A — fS 

 = PE + QF+(PE-QF)cos^ + (QE-i-PF) sin ^ 

 Expanding cos (- - A - <^V and arranging the coefficients of sin (p and 



cos <^, this may be written 



Msin<? + Ncos<^ = K (108) 



where 



M = QE + PF-2csin (^-A) (109) 



N = PE-QF-2ccos(j-A^ (110) 



K = 2ccos A-PE-QP (Ill) 



The envelope of (108) is obviously 



and this, being written in the form 



M2=(K + N)(K-N), 

 leads, on substitution from (109), (110), and (111), to the equation 



(QE + PF)2-4c sin (^- A^ (QE4-PF)+4c2 sin?' (~- a) 

 = 4c2 I cos2 A-cos2 /^-AA 



-4c |[cos A-cos ^^-A^JPE+ [cos A + cos (^-A^J QF | 



+ 4 PQEF, 

 which may be written 



