298 PARABOLIC ASYMPTOTE. 



Suppose in this case q n 2, so that equation (3) of 

 Art. 27 -i gives 



Since < (yu-J = and 0' (/U.J = 0, we have, by Art. 92, 







also 



x 



Substitute these values in the equation above, multiply by 

 a?, and then proceed to the limit, and we have for determining 

 the limiting values of ft, the quadratic equation 



If the values of ft be possible, we thus obtain two parallel 

 asymptotes. 



If this quadratic assume an indeterminate form, we may 

 proceed in the same manner to form a cubic equation in ft. 



In the case where <f>' (/it,) is zero and -\|r (/i t ) is not zero, 

 there is no rectilinear asymptote for the root yu., of the equation 

 <f) (//,) = 0, as we have already stated at the beginning of this 

 Article. In this case we may in general obtain a parabolic 

 asymptote, as we will now shew. 



By Art. 92, since $ (/*J = 0, and tj> (/^) = 0, 



1/3 2 

 : 2^ 2<3 



Hence equation (3) of Art. 274 becomes 



as a; increases indefinitely this equation approaches to the 



, 1ft 3 -drfa) L ft ( 



form ^ = - T/;7 1 \, so that = J 

 " 



