Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

21

As a functional of 56n, this expression, according to (27), is its
own adjoint; for the integral is symmetrical in M and M1, and
the term outside the integral is merely of the form of a function
of M multiplied into bin(M). Hadamard's equation is therefore
completely integrable.
On the other hand, the condition of integrability for the somewhat similar equation
(30) V'[C I ABM] = 4[C AM4[C I BM]
reduces by (27) to the condition:
4(AM)4(MM1)b(BM1) + b(AM1)b(MM1)b(BM)
= (AM1)P(MiM)4(BM) + b(AM)c(MiM)~4(BMi)
or
{ (AM)P(BM1) + ~(AM1)B(BM) } {4(MM1)
(30')
(- ~(MiM)} = 0.
In these equations the C is omitted for brevity; it must be
remembered that the points M and M1 are however restricted
to lying on the curve C.
If the first factor of (30') is identically zero, we find 4[C1 AM]
=0, by putting B = A. Hence by (30), V'[CIABM] vanishes
identically, and therefore, by (7),
D[C[ AB] = const. 0.
If on the other hand, the second factor vanishes identically
(which is the alternative if we restrict ourselves to functionals 4
which are analytic in their point arguments) we have
[Co I AB] = -[Co I BA]
provided that Co goes through both A and B. But since from
(30), for every C, A, B,
V'[CIABM] '[CIBAM],
it follows by (7) that the function 4[CI AB] - [C I BA] is independent of the curve C. Hence, identically:
4[C I AB] = [C I BA],
which is a sufficient condition for integrability.