Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

79

in which the coefficients are defined by the equations
all = an1, a22 = 22, 2a12 = a12 -+ 021,
Daa1 Da21 da12 d022
(9") a 1 +- +, a2- + + a2,
da1 da2
a + + a.
Additional derivatives would be involved in the definition of the
adjoint expression to (9'). The assumptions we shall use are,
however, given below.
With (8) and (9) we consider the expression
(10) r[C, v,] = (i { -11 D + 12 + 1 dy
Dx ay
- 21X d+ a22 + lv dx
and the equation
(11) rI[C, v] = fS (g - Ov)dxdy
which we call the adjoint of (9), provided that the relations
ij = aji, i, j = 1,2,
(12) fi = - a, i = 1, 2,
Dai D0a2
= a + + y'
9x 6y
are satisfied. If the last of these equations is written in the symmetrical form
(12') 2a + a+ 2 =2 + 3- +
it is seen at once that the relation between an equation and its
adjoint is symmetrical: if (11) is the adjoint of (9) then (9) is
the adjoint of (11).
The curve C is to be restricted to what may be called a standard
curve. A standard curve is a closed curve which does not cut