300 RECTILINEAR ASYMPTOTES. 



axis of y thus. Arrange the equation according to powers 

 of y ; suppose it to be 



?"/(*) + y n ~*A () +*"-'/.() + ... = o, 



where a, /9, ... are all positive, then the asymptotes parallel to 

 the axis of y will be given by the real roots of the equation 



For the equation to the curve may be written 



J \ I ' yO. yP 



and it is obvious that this is satisfied by supposing y oo and 

 f(x) = Q; and that when y is oo no other value of x except 

 those derived from/" (#) = will satisfy it. Hence the asymp- 

 totes parallel to the axis of y are found by equating to zero the 

 coefficient of the highest power ofy in the equation to the curve. 



Similarly the asymptotes parallel to the axis of x may be 

 found by equating to zero the coefficient of the highest power 

 of x in the equation to the curve. 



When a curve is given by a rational integral algebraical 

 equation, it will be convenient to determine by the preceding 

 method the asymptotes parallel to the axes, and then proceed 

 for the other asymptotes according to the following rule ; we 

 suppose the equation of the n ih degree. Substitute for y in 

 the given equation px + ft and arrange the terms of the equa- 

 tion according to powers of x. Equate to zero the coefficient 

 of x n ; this will give an equation for determining /j, ; suppose 

 /*, one of the real values of p. Then examine the coefficient of 

 a;"" 1 , and give to /* if it occurs in this coefficient the value yti, . 

 If we can determine /3 so as to malle this coefficient vanish, 

 then y = ^ x + /3 will be the equation to an asymptote ; if the 

 coefficient cannot be made to vanish there is no corresponding 

 asymptote. If the coefficient vanishes whatever be the value 

 of /3, then put the coefficient of x n ~* equal to zero, substituting 

 yu-j for fj, in it ; we shall thus have generally a quadratic equa- 

 tion to determine the values of /3, and if these values are real, 

 we obtain two parallel asymptotes. If the coefficient of x n ~* 

 vanishes, whatever be the value of & we must equate to zero 

 the coefficient of x n ~ 3 and so on. 



