26 CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



integral functions of the 11 th degree in the co-ordinates, while 

 u a , ti n _i, etc., or v Q , w Q , etc., shall denote homogeneous 

 functions of the degree indicated. 



Now, suppose p to be a (unrepeated) linear factor of u n , 

 say 



(2) u n =p.v n _ v 



Then, by the method of partial fractions, we can find 



(3) ^ = ~ + ^  



u n p v n _ x 



and p -f- a = is the equation of an asymptote. For, since 



u n =p.v n _ l 



u n _ x = a . v Q _ x -f / . v n _ 2 



it follows that 



(4) U n = (p + a) (»„_, + v n _ 2 ) + F n _ 2 . 



Therefore the line : p -\- a = o meets the curve : U n = o 

 only when also V n _ 2 = o, that is, in but n — 2 finite points. 

 The remaining two points of intersection are accordingly at 

 infinity, and p -f- a = o is an asymptote. 



II. Parabolic Asymptotes. Suppose u a contain a square 

 factor, say 



(5) ti Q = p" . v Q _ 2 . 



Then, applying the method of partial fractions, 



(6) 2s^ =* +!!»=? , 



where q is a linear homogeneous expression, and the par- 

 abola: p 2 -\- q = o is asymptotic to the curve. For, since 



«n = f  ^n-2 



it follows that 



(7) U*= if + ?) (^n_2 + *V-|) + ^n-2- 



The parabola: p 2 -\- q = o meets £7 n = <? only when 

 K n _ 2 =0, that is, in 2 (« — 2) points. The remaining four 

 points of intersection are at infinity. But any parabola of 



