1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 343 



so that 



iyj = h^siii^<t>cofi^<p. (173) 



Again, from (170) and (171), 



|2 rj% 



cos^ </> sin^ (p 



=za^-h^ 



or ^ sin2 <t> — r)^ cos^ <^ = c^ sin^ <f> cos^ </> = — rr- , 



from (173). 



This may be written 



|2sin2<^-7?2 (1 - sin2 (^) = p ^, 

 whence 



«-^* = FT? + ll^^' ('^^) 



Substituting for sin <^ and cos <p from (174) and (175) in (173), and 

 simplifying, we have 



which is the equation of the locus in question. Hence, we have the 

 theorem that the locus of points on a system of confocal ellipses where 

 the tangents cut off a constant area from the axes is the bicircular 

 quartic through the origin 



{c^x + h'y){h^x-c^y) = h^ (x' + y^f, (176) 



where c is half the distance between the foci, and h^ double the given 

 constant area. 



It is not difficult to see that this quartic-locus is the inverse of a 

 central conic, for, substituting for x and y 



— and „ „ 



respectively, we find that the bicircular quartic is the inverse of the conic 



(c^x-\-h^y)(h\--c^y)=h^k\ (177) 



where k is the radius of inversion ; it is easy to see that this conic is an 

 equilateral hyperbola concentric with the confocal ellipses, and, if ^ be 

 the inclination of its transverse axis to the line joining the foci of the 

 confocal family, we have 



tan 2^ 





which furnishes for tan 6 the two values 



