498 A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION [153 
which stands for 
{ax + by + cz ) x 
+ (a'x + b'y + c’z) y 
+ {a"x + b"y + c"z) z, 
and in which (x, y, z) are said to be the nearer variables, and (x, y, z) the further 
variables of the bipartite. 
2. It is clear that we have 
(a , b , c $>, y, z$x, y, z) = { a, 
a', b', c' b, 
a!", b", c" i c, 
a', a" $x, y, z$cb, y, z) 
b', b" 
and the new form on the right-hand side of the equation may also be written 
(tr. ( a , b , c ) $x, y, zjcc, y, z), 
a' , b', c' 
a", b", c" 
that is, the two sets of variables may be interchanged, provided that the matrix is 
transposed. 
3. Each set of variables may be linearly transformed: suppose that the substitu 
tions are 
and 
{x, y, z) = { l , m , 
V, ra\ 
n fe y„ *,) 
n' 
m , n 
(x, y, z) = ( 1 , 
m, 
I n, 
1', 
1" 
y, 
m', 
m" 
/ 
// 
n , 
n 
Then first substituting for (x, y, z) their values in terms of {x /t 
becomes 
a , 
[2, 
to , 
n 
a', 
6', 
c' 
m , 
n' 
a", 
b‘", 
c" 
r, 
to", 
n" 
Vo y> 
y„ z,), the bipartite 
z ) ’ 
represent for a moment this expression by 
(A , B , C $> /} y„ z&x, y, z), 
A', B', C" 
4", 5", (7'