Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

85

the equation
f If (vf - ug)dor = -H[Sr, us, v].
If, having found this equation, we now change Sr as we again
decrease or,, keeping however Sr the largest possible boundary of
squares enclosed in C, and take the limit of both sides as or
approaches zero, we have
(vf - ug)do = H[C, us, vs],
since the integrands in the curvilinear integrals constituting the
right-hand member are uniformly continuous functions of their
arguments. This completes the proof.
51. A Proof by Approximating Polynomials. An interesting method of
proof of this same theorem is afforded by the method of approximating polynomials. In what follows only a special case is treated. Let u, v and their
first partial derivatives, and f and g be limited and continuous within and on
the boundary of a rectangular region D: a x - b, a f y c5 b, and consider the theorem with special reference to Poisson's equation:
THEOREM. If for every standard curve enclosed entirely within the region D
the two equations
(20) f ds = ff f (xy)dxdy,
(21) ds =ff g(xy)dxdy
are satisfied, then the equation
(22) J (v u - u a) ds= f(vf - ug)dxdy
will also be satisfied.*
Denote by kg, the quantity
(2A)!!
k- =2
(2,s + 1)!!'
and by D' the region a' c x c b', a' c y c b', where a' and b' are fixed so
that a < a' < b' < b. We shall prove the theorem first for the region D'.
For the sake of convenience in notation we assume that 0 < a < b < 1.
* A different method of considering V2u as a single differential operator has
been developed by H. Petrini, Acta Matematica, vol. 31 (1908), see p. 181.
Green's theorem may also be proved for this operator.