RECTILINEAR ASYMPTOTES. SOI 



This rule can be easily shewn to agree with Arts. 274 

 and 275. Equation (1) of Art, 274, may be supposed the 

 equation to the curve in which n is an integer, p n 1, 

 q=n 2, ...... Then if we put px + fi for y, and arrange 



the terms according to powers of x, we shall obtain the ex- 

 pression 



Thus by equating to zero the coefficient of x n we arrive at 

 the equation for determining ju- given in Art. 274. Then by 

 equating to zero the coefficient of x n ~ l we shall obtain the 

 same value of /3 as that found in Art. 274 ; or if the coeffi- 

 cient of x"' 1 vanishes, whatever ft may be, then by equating 

 to zero the coefficient of x n ~ 3 we arrive at the quadratic equa- 

 tion given in Art. 275. 



Example (1). y 3 + x* 3axy = 0. 

 Put px+fi for y, then 



(fjus + /3) s + x 3 - Sax (px + 0) = 0; 

 therefore (/j? + I ) x s + 3x* (/*' - a/*) + Mx + N= 0. 



Hence, /a* + 1 = 0, 



a/* = 0, 



are the equations from which //, and /3 are to be found ; they 

 give /t = 1, /3 = a ; therefore 



#= x a 



is the equation to an asymptote. 



Example (2). # 2 (x + y) =<z 2 (x y). 

 Put ftx + @ for ?/, then 



a;*(a; + px + /3) = a? (x px ft) ; 

 therefore a 3 (1 + /*) + /3 2 - cca 2 (!-/*)+ 2 y3 = 0. 



Hence, 1+/A = and y9 = 0; 



therefore ^ = x is the equation to an asymptote. 



