in 2 points ; and through this a line meeting the cubic besides in 1 point ; this 
will be the before-mentioned residual point. 
The general proof is such as in the following example : 
Take on the cubic U 3 a system of 3k — 2 points, say the points a : through these 
a curve V k , besides meeting the cubic in 3k — 3« + 2 points ¡3 : and through these a 
curve besides meeting the cubic in a point G. And again through the 3k — 2 
points a a curve Wv, besides meeting the cubic in 3k' — 3k + 2 points /3' : and through 
these a curve Q k ’- K+l , besides meeting the cubic in a single point; this will be the 
point C. 
The proof consists in showing that we have a curve A 2 such that 
Aic+v-k- 2 U 3 = Qk'-K+i V k + P k _ K+ 1 W K . 
For this observe that 
Qk'- K +i meets Wk in 3k' — 3k + 2 points /3' and besides in k' 2 — k' {k + 2) + 3k - 2 points e ; 
Pk-K+i meets V k in 3k — 3k + 2 points /3 and besides in k' 2 — k (k + 2) + 3k — 2 points e ; 
P fc _ K+a , Qjc’- K +i meet in (k — k + 1) (P — k + 1) points G ; 
V k , Wu meet in 3k —2 points a and kk' — 3k+ 2 points a; 
Qk-K+i V k and P k - K+l W K meet in 
kk' — k{K — Y) — k'(K — l)+ (k — l) 2 points G 
3k' -3k+ 2 
— k! (k + 2) + 3k — 2 
3k — 3k + 2 
k 2 -k(K + 2) +3k-2 
kk' — 3* + 2 
3k — 2