330 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[3 49 
r 
I put for greater convenience 
so that X = 0, Y = 
the critic centres. 
X = - 
©! ( \X 
+ 
+V v z_ 
I2I3 V#L "l - A 0i + y 0i -f- v) 
Y = — 
Z = - 
\x 
+ 
yy 
+ 
vz 
kk \0 3 "h A 0 3 + y 0 3 + v) 
a 
Xx 
+ 
WJ 
+- 
vz 
l\ 1-2 \03 "h A 0 3 -\- y 0 3 + V/ 
0, Z = 0 are the equations of the sides of the triangle formed by 
42. Then X, Y, Z may if we please be considered as new coordinates replacing 
the original coordinates x, y, z\ the relation between the two sets being given by the 
equations last written down; the values of x, y, z in terms of X, Y, Z are given by 
the converse system 
X = 07+A X + 0~+X Y + ^+A Z ’ 
y= 
z = 
0\ + y 
1 
01 + V 
X + 
X + 
7j—-— Y + ~—Z, 
0 3 + y 0 3 + y 
Y + 
0 3 + v 0 3 + v 
Z. 
43. To show the identity of the two systems, I start 
one ; this gives 
from the last-mentioned 
02 + \’ 
1 
03 + A 
= X 
1 1 1 
01 + A 0 3 + A 0 3 + A 
^ 02 + fl’ 
1 
03 + y 
111 
01 + + 02 + y* 0 3 + y 
1 
1 
111 
0 2 + V’ 
0 3 + V 
01 + V ’ 02 + V ’ 03+ V 
where the coefficient of X is 
_ _ _ (y -v)(v- A) (A - y) (0 2 - 0 3 ) (0 S - fl) (0x - 0 3 ) 
(0i + A) (0 1 + y) (0 1 + v) (02 + A) (02 + y ) (02 + v) (0 3 + A) (# 3 + y) (0 3 + v) ’ 
or, what is the same thing, 
_ (y — v) (v — A) (A — y) 3 
©1@ 2 ©3 
The first side is a linear function of x, y, z which vanishes for 
x : y '■ z = 0^ : 0^ : faYTv’