Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

11

due respectively to small displacements of area o- and O2, of an
element ds of the curve, in the two planes. Consider each Vi as
directed normally to its plane in such a way as to make the
direction around -i positive, looking down this normal. Let V
be the vector perpendicular to C whose components are V,
and V2.
If V[CI M] is continuous in C and M in the neighborhood of
the point and curve in question, we can deduce in the same way
as for the corresponding theorem about directional derivatives
in the differential calculus, that the rate of change of F[C] for
a displacement a in any plane containing the curve is the component of V perpendicular to that plane.
From the fact that, except for infinitesimals of higher order,
we have the equation:
AF = Vndo = (X6x + Y6y + Zbz)ds
we may deduce the equations
X = Vz cos s, y - Vy cos s, z,
(18) Y = Vx cos s, z - Vz cos s, x,
Z = Vy cos s, X - Vx cos s, y,
which may be expressed in the shorter form
(18') R = r X V,
where r is a unit vector in the direction of the curve, and r X V
stands for the vector product of the two vectors r and V (the
vector area of the parallelogram of which they are the two sides).
In fact if the element of arc ds has an arbitrary vector displacement 8p we shall have
dao = (5p X r)ds
and
AF = V.do = V.(6p X r)ds,
where V da stands for the scalar product of the vectors V and
da.* But V. (p X r) = 5p (r X V), and therefore, since
* If a and,3 are two vectors, a- f is defined as a1xj + cayfy + -az3z. The
quantity a c (,3 X ) is thus seen to be the volume of the parallelopiped of