482 PROCEEDINGS OF THE AMERICAN ACADEMY. 



2y = fi v {jx v Qi + v - 2) 2 + 2 [kk' (k - 1) (k> - 1) - (2 1 + 3 /? + 5)] 



(jjl + v - 2) + 8} + {££' (& - 1) (V - 1) 

 - (2< + 3/3 + o)} 2 -(3/3+ 25), 



2x = /xv {^v(> + v-2) 2 +2[££'(£-l) (#_l)_(2* + 30+2)] 

 (ji + v-2)} + {kV (i-1) (V-l)-(2t+ S/8+2)}«- 08 + 4), 



2^ = /x v {9 <u v (/x + v - 3) 2 + 2 O + v - 3) [9 kk' (k - 1) (k> - 1) 

 -3 (6<-8/?)]} + {3 ££'(&- 1) (k'~ l) _ (6< + 8/?)} 2 

 - 22 {/x v (/t + i/ - 3) + k k' (k — 1) (V — 1) — 2 < — 3 £} 



+ 5(«v-2/3); 



where r, w, a, y, x, and ^ are the rank, class, number of stationary planes, 

 number of planes through two lines, number of points on two lines, and 

 number of lines in two planes, respectively, of the system. 



4. If the intersection of the two surfaces U and V breaks up into two 

 components of orders m and ml, we have p. v = m + m'. If the point 

 (0, 0, 0, 1) is a p-tuple point on the component of order m and a p'-tuple 

 point on the component of order m', we have p + p' = kk'. We shall 

 represent the number of lines through two points on the component of 

 order m by h, those through two points on the component of order m' by 

 h', and those through a point on each component by H". Considering 

 the intersection of the first component with the surface S, we have 



m 0* - 1) (y — 1) = 2 h + H" + P (k — 1) (V - 1). 

 Similarly, for the other component we have 



m' Qi - 1) (y - 1) = 2 h> + H" + p i (k - 1) (k> - 1). 



From these, by addition and subtraction, we obtain 



p,v (p - 1) (v - 1) = 2 (h + h! - H") + kk' (k - 1) (if - 1) ; 



and (m - m') (p - 1) (v - 1) = 2 (h - h') + ( p - P ') (k - 1) (V - 1). 



Therefore, summing for all points that are &-tuple points on U and 

 &'-tuple points on V, we have, if the curve is the complete intersection, 



h = * [p ( M - i) (* - 1) - ^ ** (* - i) (*' - i)] ; (i) 



or, if the curve breaks up into two components of orders m and m! that 

 contain the points as p-tuple and p'-tuple points, respectively, 



