PROBLEMS AND SOLUTIONS. 
598 
[383 
and if {x x , y x> z^ are the coordinates of any other point on the same cubic curve, then 
we shall have 
x x 3 + y* + ¿i 3 _ ^ _x 3 + y 3 + z 3 
XiViZi ~ ~ xyz ' 
so that {x x , y x , z x ) will satisfy the condition in question. Hence, if from a given point 
{sc, y, z) on the cubic curve we obtain by any geometrical construction another point 
on the curve, the coordinates of this new point will be functions (and, if the construction 
is such as to lead to a single point only, they will be rational functions) of {x, y, z), 
satisfying the condition in question. 
For instance, if the point (x, y, z) be joined with any point {a, /3, 7) on the curve, 
the joining line will again meet the curve in a single point, which may be taken to 
be the point (x 1) y x , z x ). This assumes that we know on the cubic curve a point 
(a, /3, 7); but such a point at once presents itself, viz., we may write (a, /3, 7) =(1, — 1, 0); 
which gives only the self-evident solution (x X) y x , z 1 )=(y, x, z). The point (1, —1, 0) 
is clearly one of the nine points of inflexion of the cubic curve, and by using these 
in any manner whatever, viz., joining the point (x, y, z) with any point of inflexion, 
and then the new point with any other point of inflexion, and so on indefinitely, we 
obtain in connexion with the given point {x, y, z) seventeen other points on the curve, 
in all a system of eighteen points: these are 
{ x > y, z), { sc, (oy, orz), { x, w-y, &) z) {x, z, y), { x, (o z, ary), { x, orz, w y) 
(y> 2 > x \ {uy, orz, x), {w-y, wz, x) {z, y, x), (a) z, w-y, x), {arz, wy, x) 
{z, x, y), {orz, x, coy), (co z, x, oo 2 y) (y, x, z), {ary, x, w z), {a)y, x, orz) 
possessing remarkable geometrical properties; and of course each of the seventeen new 
points furnishes a (self-evident) solution of the given identity. 
But we may take (a, /3, y) = {x, y, z); the point {x x , y x , z x ) is here the point of 
intersection of the cubic by the tangent at the point {x, y, z); or say it is the 
“tangential” of the point {x, y, z). The values thus obtained for {x x , y x , z x ) are 
( x i, yi, 2 i) = {sc{y 3 -2 3 ), y{z 3 -x 3 ), z{sc 3 -y 3 )}, 
Avhich (excluding the above-mentioned self-evident solutions) is in fact the most simple 
solution of the proposed identity. In order to verify that the last-mentioned values 
of (sc x , y x , z x ) are in fact the coordinates of the tangential of {x, y, z), I observe that 
this will be the case if only we have 
{x- + 2lyz) x x + {y- + 2Izx) y x + {z- + 2Ixy) z x = 0, x x 3 + y 3 + z x 3 + Qlx 1 y 1 z 1 = 0, 
the first of which is obviously satisfied by the values in question ; and for the verification 
of the second equation, 
x \ + yi' + ■S'l 3 = x 3 {y 3 — z 3 ) 3 + y 3 {z 3 — X 3 ) 3 + z 3 {x 3 — y 3 ) 3 , 
= — x a {y 3 — z 3 ) — y 9 {z 3 — X?) — z 9 {x 3 — y 3 ), 
= {x 3 + y 3 4- z 3 ) {y 3 — z 3 ) {z 3 — ir 3 ) {x 3 — y 3 ), 
X \y x z x = xyz {y 3 — z 3 ) {z 3 — x 3 ) {sc 3 — y 3 ),