212 
[924 
924. 
ON THE NON-EXISTENCE OF A SPECIAL GROUP OF POINTS. 
[From the Messenger of Mathematics, voi. xxi. (1892), pp. 132, 133.] 
It is well known that, taking in a plane any eight points, every cubic through 
these passes through a determinate ninth point. It is interesting to show that there 
is no system of seven points such that every cubic through these passes through a 
determinate eighth point. 
Assuming such a system : first, no three of the points can be in a line : for, if 
they were, then among the cubics through the seven points we have the line 
through the three points and an arbitrary conic through the remaining four points, 
and these composite cubics have no common eighth point of intersection. 
Secondly, no six of the points can be on a conic : for, if they were, then among 
the cubics through the seven points we have the conic through the six points and 
an arbitrary line through the remaining point, and these composite cubics have no 
common eighth point of intersection. 
Taking now the points to be 1, 2, 3, 4, 5, 6, 7 ; among the cubics through 
these, we have the composite cubics {A, P), (B, Q), (C, R), where A, B, C are the 
lines 67, 75, 56, and P, Q, R the conics 12345, 12346, 12347 respectively; by what 
precedes, the points 5, 6, 7 do not lie on a line, and the points (6, 7), (7, 5) and 
(5, 6) neither of them lie on the conics P, Q, R respectively. 
The common eighth point of intersection, if it exists, must be 
(A, B, C) ; (A, Q, R), (B, R, P), ((7, P, Q) ; 
(P, B, C), (Q, C, A), (R, A, P); or (P, Q, R). 
There is no point (A, B, G). 
There is no point (A, Q, R): for Q, R intersect only in the points 1, 2, 3, 4, 
no one of which lies on A ; and similarly, there is no point (B, R, P) or (C, P, Q). 
B, G intersect in 5, which is a point on P ; and thus 5 is the only point 
(P, B, G). Similarly, 6 is the only point (Q, G, A) and 7 is the only point (P, A, B). 
P, Q intersect in 1, 2, 3, 4, which are each of them on R ; hence 1, 2, 3, 4 
are the only points (P, Q, R). 
Hence the points 1, 2, 3, 4, 5, 6, 7 present themselves each once, and only 
once, among the intersections of the three cubics, and there is no common eighth point 
of intersection.