Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

43

i' = (1, in equations (17) to (20), he is able to deduce the
following theorem.
Let 4[C] = 41[C] + iP2[C] be isogenous to F[C] = F1[C]
+ iF2[C], and without singularities within a closed surface 0a.*
Then, if the values of 1[C] are known for all the closed curves which
lie on a, the functional 4[C] will be determined for all the closed
curves of the region S, enclosed by ar.
In fact, suppose there were two such functionals, and let
("[C] be their difference. The quantities Wl' dao and W2"-dawill vanish at every point of a. Hence if we apply the equation
(25) to P"[C], the right-hand member will vanish, and we shall
have
fffSkdS = 0,
where the integration is carried out over the region S. But
since k is positive and 0 is nowhere negative, 0 must be identically zero. This implies however (provided that V1 and V2 are
not collinear) that W1" and W2" vanish identically, and hence
4"[C] must vanish identically.
28. Transformation of the Variables x, y, z. Make a transformation of coordinates x = x(xyz), y = y(7yz), z = z(xyz),
whose Jacobian D = d(xyz)/O(xyz) does not vanish; denote
by K the matrix and dyadic corresponding to D:
ox Oy oz
x Oay ax
(26) K= X.y z,
Oy Oy Oy
Ox Oy Oz
Oz dOz dz
and by A the matrix of the cofactors of the elements of K, that
is, the matrix of the quantities O(yz)/O(yz), etc.
According to (6") we have A = DKc1, and according to (22),
Lecture I, any vector flux V is transformed by the formula
(27) V = A.V.
* The vector W1 is assumed to be continuous within and on the surface ar
and its components to have continuous partial derivatives of the first order
at all points inside a.