Colloquium publications.

94

THE CAMBRIDGE COLLOQUIUM.

If g is a continuous function the coefficients of (41') satisfy the
same conditions as the corresponding coefficients of (41).
We consider a region also slightly more general than in the corresponding physical problem. Denote by o0Rxl (following a notation used by W. A. Hurwitz) a
region bounded on the left by
the line x = x0, on the right by
the line x = x1, above by the
curve y = 42(x), and below by
x., the curve y = l1(x). The functions ~i and ~2 are to be continuous with their first derivatives,
___,_x_____________ ___ and are to have only a finite
FIG. 2 number of maxima and minima
in the interval under consideration. Moreover, 41(x) > ~2(x), for x x0o.
We shall investigate solutions of (41) which are continuous
with a continuous derivative in regard to y within and on the
boundary of xoR.x (called regular solutions), and take on a continuous chain of boundary values along the open contour o0rxa,
comprised by the parts x = xo, y = 2(x) and y = 41(x) of the
boundary of oRaxl. There is one and only one such solution.
Analogously, there is one and only one regular solution of the
adjoint equation (41') in the region xoaRx which takes on given
values along the contour comprised by x = x1, y = 41(x) and
y = 42(x). The proofs of the two existence theorems are similar,
and it is therefore necessary to deal only with one.
56. The Uniqueness of the Solution. Define the function
(y - aY __-y)2
(42) hal (xy x'y) = (Y. _- y) 4(x'-)
The function h0o as a function of x, y is a solution of the adjoint
of (38'), since it is seen by differentiation to be a solution of the
equation
9y2 + 9x 0.