Colloquium publications.

46

THE CAMBRIDGE COLLOQUIUM.

Div [fdF/do] 0.
This condition is satisfied since we have
Div (f d =fDiv V+ vfV.
The notation
(35) [4C] = ffdF
is used by Volterra to denote the integral (34). If the field of
integration is a closed surface, or a complete surface boundary
to a region, we have the result:
(36) ffdF = 0,
which is a generalization of Cauchy's theorem for functions of a
single complex variable.
~ 3. ISOGENOUS NON-ADDITIVE FUNCTIONALS
30. The Condition of Isogeneity. The condition of isogeneity for nonadditive functionals has already been obtained in terms of the vector flux V,
or rather, in terms of its component normal to the curve. It is desirable to
express the same condition in terms of the functional derivative vectors,
which are uniquely determined in terms of the curve and the point where
the differentiation takes place.
Denote by R[C I M], R2[C I M] the vector derivatives of FI[C] and F2[C]
respectively, and by U1[C I M] and U2[C | M] the vector derivatives of 41[C]
and ~2[C]; so that we have
(37) R = R1 + iR2, U = U1 + iU2.
If we denote by r the unit vector in the direction of the curve, and let V
and W be the normal components of the vector fluxes, we shall, as we saw in
Lecture I, have the relations
(38) R = r X V, U =r XW.
Form now the vector product, multiplying r left-handedly on to the equation
(9); we have
U2 = - (R13 - R2a) W1.
Let us denote by A' a vector perpendicular to r, such that 13 = 3' X r and
by a' a similar vector such that a = a' X r. We have
3 *W1 = 1' X r W1 = [0'rW1] = 3'rrX W = 1' * U1,
and therefore, treating a Wi in the same way,

(39)

U2 = - (R13' - R2a') U1.