316 A. Mukhopadliyay — Memoir on Plane Analytic Geometry. [No. 3, 



p = C-\'ex, 

 which is the form finally adopted by Gauss. Since x = p cos 6, we have 



C 



^~l-ecos(9' 

 which is the ordinary polar equation. If A = B = 0, we have 



p = —, 



which is the circle. The whole theory of lines of the second order may 

 be based on the form 



p=C + ex, 

 and, by means of this equation, Gauss has deduced the most complicated 

 properties of elliptic motion with remarkable ease and elegance. 

 §. 20. Eccentricity. — If we square the equation 

 pzzAx + By-hC, 

 and compare the result with the standard form 



ax^ + 2hxy-\'hy^-\-2gx + 2fy + c=z0y 

 we have, by equating coefficients, 



a_ A^-l ^_AB b B2-1 

 c "■ C2 ' c~ C2' c ~ C2 * 

 Therefore 



and 



ab-Ji'i _ (A^-l) (B2-l)-A2Bg 1-eg 



which lead to 



and this is the well-knovm equation for the eccentricity (§. 13). 



The value of the eccentricity in oblique axes may also be obtained 



from Laplace's equation ; for, if jp be the perpendicular on the directrix 



from any point on the curve 



p = Ax-\-By + 0, 



we have p = ep, 



(Ax + By-\-C) sin 0) 

 and P ~~ — 7' — 



VA2 + B2_2ABcoso)/ 



whence 



A2-f-B2-2ABcosa> 



e^= r-i (73) 



sm'^ (o ^ ' 



Now, squaring Laplace's equation, and substitutiug for p^, remembering 

 that in oblique axes 



p^ = x^-\-y^-\-2xy cos w, 



