PROBLEMS AND SOLUTIONS. 
580 
[383 
which is the equation of the cubic curve; and from this form the several above- 
mentioned results may be obtained without difficulty. 
To give an idea of the form of the curve corresponding to a given convex 
quadrangle 1234, and given position of the point 0, I suppose that 0 is situate 
within the quadrangle, for instance in the triangle B12. The mere inspection of the 
figure, and consideration of the conditions which are to be satisfied by the cubic curve, 
is enough to show that this is of the form described by Newton as anguinea cum 
ovali, viz., the oval passes through the points 3, 4, A, B, and the serpentine branch 
through the points 1, 2, C, 0, O'. But the complete discussion of the different cases 
would be somewhat laborious. 
[A geometrical investigation of the locus is given on p. 124 of Cremona’s Teoria 
Geometrica delle Curve Piane. Ed. [W. J. M.].} 
[Vol. ii. pp. 89, 90.] 
1533. (Proposed by Professor Cayley.)—If on the sides of a triangle there are 
taken three points, one on each side ; and if through the three points and the three 
vertices of the triangle there are drawn a cubic curve and a quartic curve, inter 
secting in six other points ; then there exists a quintic curve passing through each 
of the three points, and having each of the six points for a double point. 
Solution by the Proposer. 
Let P = 0 be the equation of the quartic curve, Q = 0 the equation of the cubic 
curve, M = 0 the equation of the three sides of the triangle; then if we can find 
A, B, G functions of the orders 0, 1, 2 respectively, and U a function of the fifth 
order, such that we have identically MU — AP 2 + BPQ + GQ i ; we have MU = 0, a curve 
of the eighth order, having a double point at each of the points (P = 0, Q = 0), which 
points are the three vertices of the triangle, the three points, and the six points; 
but the curve MU = 0 is made up of the curve M = 0 (the three sides of the 
triangle, being a cubic curve having each of the vertices for a double point, and 
passing through each of the three points) and of a certain quintic curve U = 0; 
hence the quintic curve must pass through each of the three points, and have a 
double point at each of the six points; or there exists a quintic curve satisfying the 
conditions of the theorem. 
I take x = 0, y = 0, 2 = 0 for the equations of the three sides of the triangle, and 
then (the constants being all of them arbitrary) writing for shortness 
£ = ■ by+ cz, 
g = a'x . + Sz, 
Ç = a'x + b"y . , 
X = . ßy + yz, 
Y = OLX . + 7 ’z, 
Z = a'x + ß"y . , 
0 = \x + gy + vz,