Colloquium publications.

26

THE CAMBRIDGE COLLOQUIUM.

By means of the definition of functional derivative, this yields:
(36) b,'[C, u M] = - 2 au(M)
an
On the other hand, if we vary C, and vary u(M) on C at the same time in
such a way that u(xy) remains unchanged inside C, that is so that 5u = au/an
8n, we have for the new quantity
5 = fc {u'(M) u(M) + Dc'(M) n(M)}ds
= j{ Ju'(M) 'n + bc'(M) n(M)ds
the formula
a42 = — w~ ) + (d) d}n ds,
~= -fr{(u ) +(u) } ends,
where u' denotes au/Os. Hence:
auu - (U\2 - 2,'(+u'(M) + ('M)= - ) -
By means of (36) this yields the equation
(37) 0c'[C, u M] = i{ '[C, u M})2 -{u(M)}2,
which is a partial equation in the functional derivatives of 1[C I u] involving
only the independent arguments C, and u(M) on C (since au/as is known when
u is known on C).
Equations of the type (37) may be called partial variational equations, or
equations in partial functional derivatives. We may expect to find them for
such functionals as are related to partial differential equations by means of
the Calculus of Variations; for some such connection is necessary in order to
eliminate the interior values of u(xy) from explicit connection with the quantities considered. Quantities like the area of minimal surfaces, the energy in a
changing system, etc., will therefore satisfy such equations. In particular,
to give another special example, if within the closed curve C, the quantity
u(xy) is a solution of the equation
a2u a2u
-+- = Xu,
ax2 ay2
the quantity
w '[CJu] {,K() a) +du )' - u2 dxdy
(C)
satisfies the partial variational equation
(38),rc'[C, u | M] = 1{,U'[C, u I M]}2 - u'(M)2 - Xu(M)2.
20. The Condition of Integrability. The equation to be considered is of
the form
(39) bc'[C, u | M = F[C, u,,bu' I,, M].