1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry, 315 



e. PM = PS, 

 whence 



But, as PM is parallel to SZ, we have 



PM =2? — X cos 6—y sin 0, 

 whicli gives 



{ep — ex cos O^ey sin B)'^-=x'^-\- t/^, 

 as might also have been obtained, but not so easily, by putting 

 sc= Xcos^ + Ysin^ 

 2/=--Xsin^ + Ycos 6 

 in the equation 



{x-¥aey y^_ 



Comparing this with the equation 



(C + A^ + B2/)2 = p2 = ^2 + 7/, 

 we get 



C = ep, A = — e cos ^, B = — e sin 6^ 

 whence, as before, 



e2 = AS + B». 



Also tan^=— Y» 



and pz= —= — 7 . 



e >x/a2 + B2 



Now, when ^ = 0, the new axis of X coincides with the major axis of the 

 ellipse ; but, when ^ = 0, we have also B = 0, by virtue of the relation 



T> 



tan 6 = — — ; 



A, 



therefore 



(C+Air)2 = ^2 + 2/2, 

 and, putting iP = 0, this gives, as before, 



Again, the equation of the directrix is 



X cos + y sin 6 =p, 

 which, by substituting for 6 and p, gives 



A^ + B^ + O = 0, 

 and this agrees with our previous result. 



It may be noticed that Gauss uses this form of the equation of a 

 conic, and calls it the " characteristic equation " (Theoria MotuSy §. 3), 

 It is easy to see that when B = 0, we have A = 6, and 



