ASYMPTOTES. 209 



Hence wo have a parabolic asymptote determined by the 

 equation 



that is 

 is, 



Next suppose jp less than w 1 . 



Then since <j>' (^,) = equation (4) of Art. 274 will not de- 

 termine* /3 ; and instead of this equation we have ultimately 

 in the manner just shewn 



If n p = 2, we obtain 



f fa) ' 



so that if ty (/LtJ and </>" (/z,) are of different signs we have two 

 possible values of /3, and therefore two parallel asymptotes 

 which are equidistant from the origin. 



If n p is not equal to 2, we have a curvilinear asymp- 

 tote determined by the equation 



276. We have assumed in Article 274, that the limit of 



'- is finite; if it be not, the limit of - will be zero, and we 

 x y 



must examine if there exists an asymptote parallel to the 

 axis of y. This can generally be easily ascertained in any 

 particular example. Or we may put the given equation in 

 the form 



and proceed as above, 



277. If a curve be given by an algebraical equation we 

 may determine the asymptotes which are parallel to the 



