ON CURVILINEAR ASYMPTOTES. 1 



BY M. W. HASKELL, Ph . D. 



Associate Professor of Mathematics, University of California. 



The treatment of curvilinear asymptotes in all our text- 

 books is extremely unsatisfactory. It is sixty years since 

 Plucker 2 called attention to the defects in Euler's discussion 

 of this subject, and stated clearly the true point of view. 

 With the exception of a single memoir by Stolz 3 , I have not 

 found any evidence that Plucker's theory has attracted the 

 attention of mathematicians, but this is doubtless due to the 

 fact that the development of projective geometry has di- 

 verted the attention of mathematicians from metrical geom- 

 etry to the undue neglect of the latter. 



That Plucker's views have not found a place in our text- 

 books is presumably owing to the lack of a simple method 

 of deriving the results in given numerical cases. For, im- 

 portant and exhaustive as is the memoir of Stolz referred to 

 above, it can not be denied that his method is inconvenient 

 in actual practice, since it depends entirely on expansion in 

 series of decreasing powers. The object of this brief article 

 is to propose a simple method of treatment, which seems to 

 me a natural sequence of Plucker's theory. This method 

 is merely an extension of the partial fractions method for 

 rectilinear asymptotes — a method that is unrivaled for sim- 

 plicity, but has never received half the attention it deserves. 



I. Ordinary rectilinear asymptotes. For this case the 

 method may be summarized as follows. A little fuller 

 treatment can be found in Edwards' Calculus. 



Let the equation of a given curve of the n th order be 



(i) 0= U n = u a -f u n _ x -f u n _ 2 + . . . +«, + »„, 



where U n , or V n , W n , etc., will be used to denote rational 



1 Read before the California Academy of Sciences, Sept. 20, 1897. 



2 Liouville' 's Journal, Vol. I, p. 229. See also the first part of Plucker's " Theorie der alge- 



braischen Curven." 



3 Mathematisc/ie Annalen, Vol. XI, p. 41. 



[ 25 ] January 31, 1898. 



