Colloquium publications.

8

THE CAMBRIDGE COLLOQUIUM.

For convenience we denote by (ul) that part of the assumption (u) which
requires uniformity with respect to the functional argument alone. From
(a) (X) (iu) can be deduced a theorem analogous to Rolle's theorem in the
differential calculus:
Let F[Lp] = F[<2] = 0, where p01 - p2 is a function which does not change
sign in the interval ab and is different from zero only in the interval a'b'.
Then there is a function (po, of the pencil determined by <pi and p2 and a value,o,
a' c ~o c b', such that F'[<po(x) I 0o] = 0.
From this theorem follows the law of the mean, in the same way as in the
differential calculus:
LAW OF THE MEAN. Let (p - (P2 not change sign in the interval ab, and be
different from zero in the sub-interval a'b' (which may be ab itself). There is a
function qoo of the pencil determined by 1p and <p2, and a value 4o, a' ' to - b',
such that
(9) F[P2(x)] - F[pi(x)] = F'[po(x) t] f: (2(x) - p(x))dx.
fa
Let us consider now functions so(x) and p(x) + -wV(x), in the given range,
w being an arbitrary parameter whose values are restricted to the neighborhood of c = 0, and make the assumptions (a) (X) (Mu). We can find an
explicit expression for [dF/dw],=o. In fact, this is easily shown to be of the
form
(10) (dF =f p(x)
of which equation (5) is an obvious consequence. Hence also we have:
(11) ( d):0 = a F'[(p(x) + c(x) I ]()d
From (11) may be deduced the equation
(12) F[p(x) + &1(x)] - F[p(x)] = I f F'[(x) + 0t(x) ] ]()d,
fa
where 0 < 0 < 1. And from (12) follows the equation (7), already given.
~ 2. FUNCTIONALS OF CURVES IN SPACE
7. Introduction. We are interested in this lecture not so
much in the character of the curve, which is the argument of the
functional, as in the character rather of the functional relation
itself. And so we shall assume without statement, or with slight
statement, whatever properties may be needed from time to
time in order to make possible the differential and geometric
transformations used in the analysis of the functionals. The
curves are to be closed, and in particular, each one must be
capable of being capped by at least two surfaces which have no