Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

23

SA[C] = J '[C |M]6n(M)ds,
which is an identity for any given functional 4[C], under the conditions described for (7). Let us suppose that 4'[C I M] has itself
an integrable derivative 4"[CI MM1]. We shall have then for
56' the expression
I '"[C I MMi]61n(M)1ds,
plus possibly other terms which are not integrals. The condition
"[C IMM1] = "[C MM]
must therefore be included in the condition of integrability, a
result which was originally stated by Volterra. The second
functional derivative must be symmetrical in its point arguments.
17. The Integrability of the Equation for the Green's Function. For the
equation
(32) '[C I ABM] = - [C AM] [C MB]
an an
which is equivalent except for a constant to (24), we have
'(ABM) C[d a4(AM) aI(MB) a2I(MM,)
5 '(ABM) -= - an an, anan
a Q(AM1) O~(MB) a'2(MM) 1,n(s,)dsj
an, an anani,
a ( a-i(AM)ad (MB) 61n(s)
+ ( 9a(AM) ad (MB) + a((MB) a~(AM ),
+ s n a s an ) 1n(),
in which the last two terms represent the variation of the right-hand member
of (32) due to the displacement of the point M, the last term arising from the
change of direction of n. The notation S6n'(s) stands for the derivative with
respect to s of the quantity 5in.
In order for this expression to be self adjoint, we have, by (27):
a(AM) ad (MB) a (MB) a9 (AM) =
(33) as~ On q s-~- On
as an as an
for all curves C, all points A and B, and all points M on C. Equation (32) is
therefore not completely integrable. Since (30) is an identity, its variation also
must vanish. This will be constituted by an integral, a term in An and a
term in Wn'. By choosing particular types of functions 6n(s) it is easily seen
that each of these terms must vanish separately.