[349 
349] 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
339 
We 
Article Nos. 53 to 55. Transformation of the Equation of the Cubic. 
53. Let it be required to express the cubic 
xyz + h (x + y + z) 2 (\x + yy + vz) = 0 
in terms of the coordinates X, Y, Z. We have 
a+ y+ z = 2(jX + ~Y + j^z), 
\x + yy -f vz — 
and the equation therefore is 
n (ft 
XYZ 
+ 7T-T— + 
■h A 02 "h ¡a 02~\~ v 
X+ Y + 
+ «l| + | + |) , (Z + r + ^) = 0. 
where II denotes the product of the three factors obtained by writing y, v 
successively in the place of 
-w 
©! ’ 
For one of the nodal cubics we have 
k = h = 
and the equation multiplied by ©j is 
n j x + T -W + -{ x+Y b4J (X+r+Z)=0 - 
which it is clear a priori must be of the form 
e^ + <àlò- e 'ÌbfJ (Y+Z) = 0 ’ 
and there is in fact no difficulty in verifying that the coefficients of X 3 , X 2 Y, X-Z, XYZ 
all of them vanish. To find K, comparing the coefficients of XY 2 we have 
that is 
JZ 1 = V (01+ x ) (01 + f) _ Of _ o 01 
' 4 S 0,®2 (0.+ X)(0 a + ^) 02 2 02 ’ 
K. A = 2 (ft + X) (ft + p)(9, + v)- (0, + 2ft) 0«, 
= 2 (ft + X) (ft 4- fi) (ft — ft + ft + v) — (ft + 2ft) © 3 , 
= j(ft - ft) I + sj ©, - (ft + 2ft) 0„ 
= g- (ft + 2ft) 0, - ^ (ft + 2ft) ®„ 
_ k b JAc, 
~ e 1 1 di 2 2> 
= g(W-W: 
43—2