153] 
OF A BIPARTITE QUADRIC FUNCTION. 
499 
then substituting for (x, y, z) their values in terms of (x,, y„ z,), it is easy to see 
that the expression becomes 
((1, m , n # A , B , G ) y t , z$x /t y„ Z/ ), 
1m', n' 
5', 
C' 
1", m", n" 
-S", 
G" 
and re-establishing the value of the auxiliary matrix, we obtain, as the final result of 
the substitutions, 
( a , b , c 
fa, y, zjx, y, z) = ( ( 
1 , m , n ] 
[a, b,c } 
[l , m , n 
a , b', c' 
1' ,• m', n' 
a , 5', c' 
V, m, 72' 
a", b", c" 
1", m", n" 
a", 6", c" 
l", m", n 
that is, the matrix of the transformed bipartite is obtained by compounding in order, 
first or furthest the transposed matrix of substitution of the further variables, next the 
matrix of the bipartite, and last or nearest the matrix of substitution of the nearer 
variables. 
4. Suppose now that it is required to find the automorphic linear transformation 
of the bipartite 
(a , b , c $ x, y, zjx, y, z), 
a , V, c 
a , b , c 
or as it will henceforward for shortness be written, 
y, z~$x, y, z); 
this may be effected by a method precisely similar to that employed by M. Hermite 
for an ordinary quadric. For this purpose write 
x + x=2%, y+y=2 v , z + z / = 2g, 
x + x / = 2E, y + y, = 2H, z + z ; = 2Z, 
or, as these equations may be represented, 
(« + »„ y + y,, z + z)= 2(|, y , £), 
(x + x,, y + y,, z + z / )=2(E, H, Z); 
then we ought to have 
(f!3[2£-ic, 2?;- y, 2£-/£2E-x, 2H-y, 2Z - z) = (ilfa, y, zjx, y, z). 
5. The left-hand side is 
4(il5f, y, r$B, H, Z)-2(il$*, y, 3, H, Z)-2(fi$f, y, y, z) + n\x, y, z$x, y, z), 
and the equation becomes 
y, 3, H, y, ¿$5, H, Z)-(fi$f, y, y, z) = 0, 
63—2