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25 
ON THE INTERSECTION OF CURVES. 
[From the Cambridge Mathematical Journal, vol. in. (1843), pp. 211—213.] 
The following theorem is quoted in a note of Chasles’ Aperçu Historique &c., Mémoires 
de Bruxelles, tom. XI. p. 149, where M. Chasles employs it in the demonstration of 
Pascal’s theorem : “ If a curve of the third order pass through eight of the points of 
intersection of two curves of the third order, it passes through the ninth point of 
intersection.” The application in question is so elegant, that it deserves to be generally 
known. Consider a hexagon inscribed in a conic section. The aggregate of three 
alternate sides may be looked upon as forming a curve of the third order, and that 
of the remaining sides, a second curve of the same order. These two intersect in nine 
points, viz. the six angular points of the hexagon, and the three points which" are 
the intersections of pairs of opposite sides. Suppose a curve of the third order passing 
through eight of these points, viz. the aggregate of the conic section passing through 
the angular points of the hexagon, and of the line joining two of the three inter 
sections of pairs of opposite sides. This passes through the ninth point, by the theorem 
of Chasles, i.e. the three intersections of pairs of opposite sides lie in the same straight 
line, (since obviously the third intersection does not lie in the conic section) ; which is 
Pascal’s theorem. 
The demonstration of the above property of curves of the third order is one 
of extreme simplicity. Let U = 0, V = 0, be the equations of two curves of the 
third order, the curve of the same order which passes through eight of their points 
of intersection (which may be considered as eight perfectly arbitrary points), and a 
ninth arbitrary point, will be perfectly determinate. Let U 0 , V 0 , be the values of 
U, V, when the coordinates of this last point are written in place of x, y. Then 
UV 0 —U 0 V=0, satisfies the above conditions, or it is the equation to the curve 
required ; but it is an equation which is satisfied by all the nine points of intersection 
of the two curves, i.e. any curve that passes through eight of these points of inter 
section, passes also through the ninth, 
c. 
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