M.-P.— Vol. I.] HASKELL— ON CURVILINEAR ASYMPTOTES. 2J 



the family 



f + q + k = *, 



where X- is any constant, has the same property. For we 

 can easily transmute (7) into the form 



V* = (/+ ? + *) (^n-2 + ^n-3) + ^n-2, 



and the same analysis holds. 



There is therefore a family of asymptotic parabolas, meet- 

 ing U n — o in four coincident points at infinity. If, how- 

 ever, k be so chosen that one of the intersections of 

 pi _|_ q _f_ k = o with W n _ a = be also at infinity, the para- 

 bola in question will have 5-point contact, or will osculate 

 U n = o. The necessary and sufficient condition for this 

 osculation is that the axis of the parabola (p = o) should 

 meet V n _ 2 at infinity, that is, that the terms of highest de- 

 gree in V n _ 2 should contain p as a factor. The value of k 

 in question can be determined by resolving 



u n _ 2 — q . v n _ 3 __ k 



■zt> 



/o\ , "n-2 y • *-n— 3 '" I ^n-3 



P . Z/ n _ 2 ^ ^ u _ 2 



For in this case 



«„_2 = k  Vn-2 + ? • ^n-3 +/  W n-3> 



and hence 



(9) cv n = (f + q + k) (v n _ 2 + ^ n _ 3 ) + / . w n _ 3 + r;_ 3 . 



The parabola thus determined will then be regarded as the 



asymptote. This distinction is not made, so far as I have 



been able to discover, except by Pliicker and Stolz in the 



memoirs referred to. 



In particular, if q is a multiple of p, say q — fip, we have 



the case of parallel asymptotes. The given curve has a 



double point at infinity on the line: p = o, and the tangents 



at this double point, or the asymptotes, are given by the 



equation : 



p 2 + fJLp -\- k — o. 



III. Cubic Asymptotes. Suppose u n contain a cubic 

 factor, say 

 (10) u n =p 3 .v n _ 3 . 



