Multiplying determinants according to the rule in § 1, we have 
a 2 / 
+ a 92 / 
dv 
dw 
a* 2 
a*ay 
dx 2 
a* ay 
dx 
dx 
a 2 / 
a 
+y% 
+e e2 { 
dv 
a w 
a*ay 
dy 2 
dxdy 
ay 2 
ay 
ay 
where, by T, 
(1) v= a 3f+y3f=Zf^ + ZfZy = ZF 
d* ay dx as ay a£ as’ 
w = /8-^ + 5- 
dx dy 
af 
dv 
By the same rule of multiplication of determinants, 
\ dx dy, 
,af 
dv 
dv 
Applying (1) with/ replaced by dF/dk for the first column 
and by dF/dv for the second column, we get 
a 2 f 
a 2 f 
de 
dtdv 
a 2 f 
d 2 F 
dvdt 
dv 2 
Hence the Hessian of the transformed function F equals the 
product of the Hessian h of the given function / by the square 
of the determinant of the linear transformation. Conse 
quently, h is called a covariant of index 2 of /. 
For an interpretation of h= 0, see Exs. 4, 5, § 7. In case 
/ is the quadratic function / of § 4, h reduces to 4d, where d 
is the invariant ac—h 2 . 
The Junctional determinant or Jacobian (named after C. 
G. J. Jacobi) of two functions f{x, y) and g(x, y) is defined 
to be 
Let 
g b 
Hei 
I a 
l ai 
and 
tral 
if 
let 
con 
case 
Let 
and 
and