Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

81

are satisfied, then the equation
(14) H[C, u, v] = fj (vf - ug)dxdy
will also be satisfied.*
This theorem contains as a special case the usual Green's
theorem for the differential expression (9'). If equation (9) is
to be investigated, the function g(xy) in (11) may be regarded
as arbitrary, and will be chosen so as best to make the equation
(9) integrable, in order that a convenient function v(xy) may
then be utilized in connection with (9) and (14).t If g(xy) is
taken identically null, the function v(xy) is an integrating multiplier for the equation (9).
50. Proof of Green's Theorem. In order to prove this theorem we follow a method worked out by C. A. Epperson,1 the
integro-differential expressions being now however somewhat
simplified in comparison with his, and establish first a lemma.
For this purpose the curves C are restricted to small squares
about arbitrary points P as center, which are made to approach
P as a limit; for convenience, the values of a function at P and
at a point on C are designated by subscripts P and s respectively.
The lemma follows:
LEMMA. With the above restrictions upon the curves C, the
relations
(15) (f au)p = lim -A[C, us],
or=P 0'
(16) (g- 3v)p = lim -F[C, vs]
0T=P O'
* Special cases of this form of Green's theorem were proved independently
by C. W. Oseen, Rendiconti del Circolo Matematico di Palermo, vol. 38 (1914),
pp. 167-179; and G. C. Evans, Transactions of the American Mathematical
Society, vol. 15 (1914), pp. 477-496. A similar theorem has been proved for
hydrodynamics by C. W. Oseen (see the footnote to the title of Lecture IV).
The theorem also holds if the curves C are restricted to rectangles or even
to circles (see the second method of proof).
t Such for instance as the Parametrix, defined by Hilbert.
+ Epperson, Bulletin of the American Mathematical Society, vol. 22 (1915),
pp. 17-26. In the demonstration referred to there is an error in the formula
corresponding to (18) below.