Colloquium publications.

22

THE CAMBRIDGE COLLOQUIUM.

Consider as a third equation
(31) b[C] = ~f(D, M)5n(M)ds,
for which we have
]Of(f, M) Of(, M)
^6@[CW = d^Sc^ 6f l+ - n^f 6n(,M)
= -f(,n ) f(b, mil),n(Mi)dsi + f ln(M).
By (27) the condition of integrability is the following:
I(31') t 3af( c, M1) = c Mf()i, M).
If the equation is to be completely integrable, this condition
must be satisfied for all functionals 4 and points M and M1 on
C. This is the same as saying that 0 logf(4M)/Od shall be
independent of M, M on C. Hence, for M on C, f must be of
the form:
f(b, M) = g(P)p(M),
and since the function does not involve C it must have this form
always. Therefore, by (30)
g() -= f(M)6nds.
The right-hand member of this equation is merely 6T[C],
where
T[C] = ffd p(M)do
(c)
is an arbitrary additive functional. But the relation
l/g(b)= at
merely tells us that the quantities 4 and T are functionally dependent, i. e., that d4/dT exists and is given by g(4). Hence
the functionals defined by (30), if (30) is completely integrable, are
merely functions of additive functionals of curves C.
As a last example, consider the equation