CURVILINEAR ASYMPTOTES. 295 



is the equation to a curve which is an asymptote to the 

 former. So also is 



y = A x n + A 1 x n ~ 1 + ...+A n _ l x + A n + - ', 



x 



7? 7? 



and i / = A x n + A 1 x n - l + ...+A a _ 1 x + A n + - + -?, 



X *C 



and so on. 



Example. Find asymptotes to the curve 

 a? - acy* + ay* = 0. 



Here tf=- \ therefore y = + 

 x-a' ~ 



x - a 



As x approaches the value o, both y and -^- increase 



without limit, and x = a is the equation to a rectilinear 

 asymptote. 



Putting y in the form oc(l -- ) , and expanding by 



\ *c/ 



the Binomial Theorem, we have 



Hence y= *+a are * ne equations to two rectilinear 



asymptotes. We may obtain as many curvilinear asymptotes 

 as we please by making use of the series in (1). For example, 



-are the equations to two asymptotic curves of the second 

 order. The student will remember that by Art. 114 we 

 may use the Binomial Theorem in the above Example as a 



true arithmetical expansion when - is less than unity, which 



C 



will certainly be the case when as is increased indefinitely. 



