Colloquium publications.

84

THE CAMBRIDGE COLLOQUIUM.

which is
/ a u du Ou O v ai
(v8' -/2 ax Qy-1 ax yJp \v ax a p
In a precisely similar manner we find for the limit of the expression (iii) the quantity
ov au Ou O v\ a a2
- a21- - a12 - -) + I U )
-- a 2 O y X Ody ax p u 9y jp
Hence the sum of (i), (ii) and (iii) has the limit zero, and the
lemma is proved.
We see that all the limits involved in the proof of the lemma
just given are uniform with respect to the point P. Hence we
have the further lemma that the limit specified in equation (17)
is uniform with respect to the point P, for all points P in a region
enclosed by any standard curve C lying wholly within A, and therefore that
(19) W( - ug)p - H[C, us, Vs] ra
where rj is independent of P, and C is the boundary of the square a.
If now we return to the original theorem, the proof is immediate. For if we divide up the region a, bounded by the
curve C, by a square grating, each square being of size ar, and
denote by Sr the outside boundary of the collection of squares
entirely enclosed within a, we shall have
f (vf - ug)da - (vf - ug)pOr cE< r,
where P is the center of the square ar, and e can be made uniformly as small as we please with ar, for all squares ar. Hence
we have the equation
f J (vf - ug)da - H[Sr, us, vS] -- (77 + e) z r,
and, since we can make the grating as minute as we please, by
further subdivision, without changing the outside boundary Sr,