PROCEEDINGS OF THE AMERICAN ACADEMY. 



No general proof of this theorem has yet been found, and it must be 

 proved for each of the ruled surfaces, separately. 



From Theorem I are readily deduced the other two theorems, as 

 follows : 



Theorem II. — If aj is the complete intersection of 2< M ) and SH"\ and 

 if bp is any curve on 2^) that has no component in common with a a , then 

 , bp) = u ft + b a — /x aft. 



Proof. — The intersections of a a and bp are simply the intersections 

 of £('•> and bp and are in uuml>er equal to b v, i. e. (a a , bp) = b v. Now 

 since each generator meets £('> in v points, a = v, also a = p v = ti a, 

 and we have 



a ft -\- b a — pa ft = fxaft + bv — fi a ft = b v; 

 therefore (a a , bp) = a ft + b a — ll a ft. 



Theorem III. — If a a is irreducible and the partial intersection of 

 2W and £<"), a' a ' being the residual intersection, and if the formula holds 

 for each irreducible component of a' a > with an arbitrary curve bp on 2 ( ^, 

 it also holds for a a with bp. 



Proof. — The residual a' > may break up into several curves, but 

 bp, being arbitrary, does not in general contain any part of the inter- 

 section of 2<^) and £H"\ If a' a > is reducible, the order a' is the sum of 

 the orders of the component curves, and the number of points a in which 

 any generator meets a 1 a < is the sum of the numbers of points in which this 

 generator meets the component curves. Since the complete intersection 

 of 2M and £(') is a a + a a > we have, by Theorem II, 



(a a + a' a , bp) = (a+a')ft+b(a + a')- t i(a + a') ft. 



Ly supposition 



(a'„ bp) = a' ft + I a - fxu ft. 



Now the number of points in which bp meets the complete intersection 

 lesa the number in which it meets a' a > must be the number of points in 

 which it meets '/„ ; therefore 



( "a, bp) = a ft + b a — f L a ft. 



Corollary. — If the complete intersection of 2' M and X" ' consists of 

 two curves and the formula holds for <>ne of these curves it holds for the 



other also. 



1. In Order then to prove the formula for any 2 (l1 it snllices first to 



prove Theorem I. and then to show thai everj curve on 2"" can be cut 



out by an & '"■ ach that the residual i.-, a curve, or is composed of CUrvt . 



