ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
349] 
and for 
1 1 1 
x '■ y '■ z ~e 3 +x '■ e,+n : 0, + v’’ 
and hence it is of the form 
fXx uy vz 
\0j + A 0 1 + /x 6 1 + v 
and by comparing the coefficients of x we have 
A, 1 1 
331 
K 
(02 - 0g) (+ - V ) 
that is 
0i + ^ (02 + +) (03 + v) (0 2 + v) (9 3 + /a) (0 2 + /x) (6. 2 + v) (9 3 + fi) (0 3 + v) ’ 
_ Zi (/x — v) ($ 2 + A) (0 3 + A) 
© 2 © 3 ’ 
7r _h(+~ V ) (01 + ^-) (02 + A) (0 3 + A) 
A©*©., 
and it is easy to see that 
($i + A) (d 2 + A) (0 3 + A) = — A (v — A) (A — fi), 
so that 
y _ - ¿i (/* ~ v) (v- A)(A-/x) 
©.©« 
and the equation becomes 
X = 
— © x / \x 
+ 
+ 
vz 
IJj 3 \01 + A 62 + fL 0x+vJ ’ 
which is right ; and similarly for the values of Y and Z. 
44. The equation of the tangents at the node corresponding to the root 9 X is 
(y + 4A, 0, + in, 0, + iv, -V.- 2 ^, -0.-^, V, ■s) 3 = 0 ; 
and substituting for x, y, z their values in terms of X, Y, Z, it appears in the first 
place that the coefficients of X 2 , XY, XZ, YZ, all of them vanish. 
45. In fact 
coeff. X 2 = I 0j + 4A, . ,. , — 9 X — 
2 fjbv 
0i ’ ’ } (0i + A ’ 02 +fi’ 0 1 + v ) ’ 
9 + -? v 
ft + 4X_ 2S "‘ + 0, 
First term is 
(0, + W ■(/>: I /,>«>: 
= £^4^ + 32 X 
0! + A (0i + A) 2 ’ 
2 3 
= 02 + ©T ~ ^ v + ^ + V)}> 
42—2