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



Then, proceeding as before, 



(II) ^"# + ^' 



Every cubic curve : / 3 -fz> 2 -f-#'-f->£ = 0, where q is an ar- 

 bitrary linear function and k an arbitrary constant, will 

 meet the given curve in six points at infinity. But if q be 

 determined by the resolution of 



Z 7 • <-n-3 .? ^n-3 



then every such cubic curve, where k alone is arbitrary, 

 will meet the given curve in eight coincident points at infin- 

 itv. For we shall have 



u Q = f. v n _ z 



«n-l = ^2 • ^n-3 + ^- ^n-4 



«„-2 = ? • ^n-3 + ^2 • ^„-4 + f' ^n-4 



and consequently 



(13) U n = (f + V a + ? + *) (»_ + *„_«) +#• *V-4 + K-Z 



There will in general be no particular one of this family 

 of cubics having closer than 8-point contact with the given 

 curve. But if v 2 contain^, say v 2 =fi.r , there is 9-point 

 contact with all cubics of the family, and there will be one 

 member of the family having 10-point contact, namely that 

 one for which k is determined by the resolution of 



"n-3 9 ' V u-i k . W 'n- 



i 



V 



u-3 



For then we shall find that 



(15) U n = {f + f r + q + k) (v n _ 3 + z< u _ 4 ) -f f. zu n _, 



+ f • w'n-4 + K-4 • 



The extension of the method to higher cases is so obvious 

 that I will not carry it further, especially since the number 

 of special cases to be considered soon becomes inconveni- 

 ently large, and it does not seem worth while to discuss 

 them. 



