383] 
PROBLEMS AND SOLUTIONS. 
599 
[383 
383] PROBLEMS AND SOLUTIONS. 599 
ve, then 
therefore 
x \ + l Jl + z l + ^lx 1 y 1 z 1 — (x 3 + y 3 + z 3 + Qlxyz) {y 3 — z 3 ) (z 3 — x 3 ) (x 3 — y 3 ), = 0 
m point 
îr point 
¡traction 
z \ 
if x 3 + y 3 + z 3 + 6Ixyz — 0 ; the same equations verify at once the identity 
X 3 + y 3 + Zj 3 X' + y 3 + z 3 
shyiZy xyz 
Another solution is as follows: viz., if we take the third intersection with the 
e curve, 
aken to 
a point 
-1, 0); 
-1, 0) 
g these 
îflexion, 
ely, we 
; curve, 
cubic of the line joining the points (y, x, z) and [x (y 3 — z 3 ), y{z 3 — x 3 ), z(x 3 — y 3 )), the 
coordinates of the line in question are 
X\ : 2/i : Zi = x 6 y 3 + y s z 3 + z e x? — 3x?y 3 z 3 
: x 3 y 3 + y 3 z 8 + z 3 x 3 — %x 3 y 3 z 3 
: xyz (x s + y 6 + z 6 — y 3 z 3 — z 3 x 3 — x 3 y 3 ). 
According to a very beautiful theorem of Professor Sylvester’s in relation to the theory of 
cubic curves, the coordinates of a point which depends linearly on a given point of the 
curve are necessarily rational and integral functions of a square degree of the coordi- 
™y) 
, x) 
nates (x, y, z) of the given point; and moreover that (considering as one solution those 
which can be derived from each other by a mere permutation of the coordinates, or 
change of x into cox, &c.), there is only one solution of a given square degree m 2 ; the 
arz ) 
solutions of the degrees 4 and 9 are given above. The tangential of the tangential, or 
in new 
second tangential of the point (x, y, z), gives the solution of the degree 16; joining 
this second tangential with the original point (x, y, z), we have the solution of the 
degree 25 ; and the same solution is also given as the sixth point of intersection with 
oint of 
is the 
the cubic, of the conic of 5-pointic intersection at the point (x, y, z). See my memoir 
“ On the conic of 5-pointic contact at any point of a plane curve,” Phil. Trans, vol. cxlix. 
(1859), pp. 371—400, [261]. 
simple 
values 
e that 
Addition to the foregoing Solution. On a system of Eighteen Points on a Cubic Curve. 
Considering the cubic curve x 3 + y 3 + z 3 + Qlxyz = 0, we have the nine points of 
inflexion, which I represent as follows: 
ication 
a = (0, 1, -1), d = (-l , 0, 1), g = (1, -1,0), 
b = (0, 1, - 0)), e —(— co, 0, 1), h = (1, -co, 0), 
c = (0, 1, - co 2 ), f = (- o> 2 , 0, 1), % = (1, - co 3 , 0), 
viz., co being an imaginary cube root of unity, the coordinates of a are (0, 1, — 1), those 
of b, (0, 1, — co), &c.