214 
ON RESIDUATION IN REGARD TO A CUBIC CURVE. 
[589 
and through these a line meeting the cubic besides in a point (7; then this is a 
fixed point, independent of the particular quartic. And the proof is as follows : we have 
U a cubic through 10 points a ; 
V a quartic through the 10 points, and besides meeting the cubic in 2 points A ; 
W a quartic through the 10 points, and besides meeting the cubic in 2 points /3' ; 
P the line joining the two points /3, and besides meeting V in two points e; 
Q the line joining the two points /3', and besides meeting W in two points e ; 
P, Q meet in the point (7; 
U, V meet in the 10 points a, and besides in 6 points a ; 
A a conic through 5 of the points a. 
Then quintics QV, PW meet in the 10 points a, 2 points /3, 2 points e, 2 points /3', 
2 points e, 6 points a and 1 point (7. Every quintic through 19 of these passes 
through the 25. But we have AU, a quintic through 5 points a, and the 10 points a, 
2 points ¡3 and 2 points A ; hence A U passes through all the remaining points, or we 
have 
au=qv+pw, 
P passes through /3, /3 , e , e , C. 
Q 
)) 
/3', ¡3', e' , e' , C, 
V 
» 
e , e , /3, A , 
6 points a, 10 points a, 
w 
>> 
e , e, A', A', 
6 points a, 10 points a, 
A 
e, e , € , e , 
6 points a, 
u 
A, A, A', A', C, 
the same 
thing, 
A, P 
intersect in e , e , 
A, Q 
/ / 
„ e , e , 
A, V 
„ e > e , 
6 points a, 
A, W 
» ^ ^ j 
6 points a, 
U, P 
» A, A, 
G, 
U, Q 
„ A', A', 
G, 
U, V 
„ A, A , 
10 points a, 
u, w 
A', A', 
10 points a. 
In particular U, P, Q intersect in the point C; that is, C as given by the inter 
section of U by the line P; and as given by the intersection of U by the line Q; 
is one and the same point.