22 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(^) and S^''\ and 

 if h^ is any curve on 2^^) that has no component in common with aa.-, then 

 (fta, bp) = a, ft + b a — ixa fS. 



Proof. — The intersections of tta and b^ are simply the intersections 

 of /S(^' and bp and are in number equal to b v, i. e. (ua, bp) = 6 v. Now 

 since each generator meets S^"^ in v points, a = v, also a == yx v = /u. a, 

 and we have 



a (3 -{- b a — fia/S^ixafi + bv — /x a ft = b v; 



therefore (Oa, h^) = a(3 + ba — fxa/3. 



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

 S^*^) and *$("), a'a' being the residual intersection, and if the formula holds 

 for each irreducible component of a'a' with an arbitrary curve b^ on S''^^ 

 it also holds for a^ with bp. 



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

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

 section of 2''*) and »S'(''). 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'a' is the sum of the numbers of points in which this 

 generator meets the component curves. Since the complete intersection 

 of S^*^) and 5('') is aa + a'a' we have, by Theorem II, 



(a„ + a'a , bp) = (a + a') (3 + b(a-h a')- fx {a + a') (3. 



By supposition 



{a'a, bp) = a' ft-\-ba' -fia' /?. 



Now the number of points in which b^ meets the complete intersection 

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

 which it meets «„ ; therefore 



(«a, ^^) = a ft -{■ b a — fxaft. 



Corollary. — If the complete intersection of S^*^) and S(^) consists of 

 two curves and the formula holds for one of these curves it holds for the 

 other also. 



4. In order then to prove the formula for any ^i") it suffices first to 

 prove Theorem I, and then to show that every curve on S^*^) can be cut 

 out by an »S('') such that the residual is a curve, or is composed of curves, 



