616] A GEOMETRICAL ILLUSTRATION OF THE CUBIC TRANSFORMATION. 
523 
Taking « an imaginary cube root of unity, write 
mx + o)-y — 21 z = x' y 
mrx + my — 2lz — y', 
x + y — 2lz = z ; 
then we have 
Also 
'y’z' = x s + y 3 — 8l 3 z 3 + 6lxyz = a? +y s + z 3 + 6lxyz — (1 + 8I s ) z s . 
Glz = x + y’ + z', z 3 = (a + y' + zj, 
whence 
so that, putting 
216£ 3 
2161 s 
216 I s 
O' + y + zj - x'y'z' = (x 3 + y 3 + ^ + 6lxyz) ; 
-I s 
or, what is the same thing, 
the equation of the curve is 
and if we write 
m " 1 + 8 I s ’ 
8 l s m s + I s + m 3 = 0, 
(pc' + y + z') 3 + 216m 3 x'y'z' = 0 ; 
x' : y' : z' = X 3 : Y a : Z 3 , 
then the original curve is transformed into 
(X s + Y 3 + Z 3 ) 3 + 216w 3 X 3 F 3 Z 3 = 0, 
a curve of the ninth order breaking up into three cubic curves, one of which is 
X 3 +Y s + Z 3 + 6mXYZ = 0, 
and for the other two we write herein mm and mm- respectively in place 
Attending only to the first curve, we have 
x 3 + y 3 + z 3 + 6 Ixyz = 0, 
X 3 + Y 3 + Z 3 + 6 mX YZ = 0, 
as corresponding curves, the corresponding points being connected by the relation 
mx + m-y — 2Iz : m-x + my — 2lz : x + y — 2Iz = X 3 : Y 3 : Z 3 , 
or, for convenience, we may write 
mx + wry — 2Iz — X 3 , giving 3x = m-X 3 + mY 3 + Z 
m~x + my — 2 Iz = Y 3 , 3 y — mX 3 + m 2 Y 3 + Z 3 , 
x+ y — 2lz = Z 3 , —6lz= X s + F 3 + Z 3 . 
of m. 
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