Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

13

construction familiar in the case of hydrodynamics shows us
that V, is merely the component in the direction N of the
vector (V., Vu, V,).
On account of (20'), Volterraluses the symbols aF/d(yz),
OF/d(zx), OF/O(xy) to denote the three components of V, and
the symbol dF/do to denote the vector V itself.
10. The Condition of Integrability for Additive Functionals.
For any closed surface we must have the equation
(21) ff Vda = 0,
since it may be regarded as forming a double cap for a closed
curve lying on it. Hence at every point in the region we are
considering, the relation
0v1 0v, OVz
(21') O x +y z
-3+- y + =
must hold; the divergence of V must everywhere vanish.
On the other hand, it is evident, that if the relation (21') holds
everywhere for the vector point function V, it will define an
additive functional of space curves by (19), of which V will be
the vector flux. Equation (21) will hold in fact for any closed
surface, and V will satisfy (20').
11. Change of Variable. If we make a one-one point transformation of space x = x(xz2), y = y(xz), z = z(xy), where
x(xyz), y(xyz), z(xyz) are continuous functions with continuous
first derivatives, with [a(xyz)/a(xyz)] + 0, the closed curve C
will go over into a closed curve C', and the additive functional
F[C] will go over into an additive functional F[C], which will
have a flux vector V.
If (u, v) are the curvilinear co-ordinates of corresponding
points on the caps 2 and 2 of C and C respectively, then
F[C] = F[C] = f fVxdydz + Vydzdx + Vzdxdy
VX d(yz) + V x) V (xy) dudv.
I (uV) a (uV) 9(U)