296 RECTILINEAR ASYMPTOTES. 



274. The following method will furnish the rectilinear 

 asymptotes with great readiness in many cases. Suppose 

 the equation to a curve, F(x, y} = 0, to be such that F(x, y) 

 is the sum of different homogeneous functions of x and y, so 

 that the equation may be put in the form 



ft] +x q x (-} + ... =0 ......... (1), 



\xj ^ \xj 



where n, p, q, are arranged in descending order of magnitude. 

 For example, every rational integral algebraical equation 

 between x and y can be put in this form. From (1) we have 



x \x/ x 



Now in finding an asymptote we must first by Art. 271 



y 



ascertain the limit of when x and y are infinite. If we 

 x 



call that limit p,, and suppose it to be finite, we have from (2) 



- 0. 



Let /*, be a value of p, obtained from this equation; we 

 have next to find the limit of y ^x. Put y ^x = /3, 

 then from (2) 



But, by Art. 92, 



since <f> (JJL^) = 0. 



Thus (3) becomes 



