Colloquium publications.

96

THE CAMBRIDGE COLLOQUIUM.

namely, the known solution of the equation
02u du
(46) ay2 0.
In particular, (46) will have a Green's function G(xy I x'y').
The function
u(x'y') = u(x'y') + - G(xy x'y')f(xy)dxdy,
where i(x'y') is the solution of (46) which takes on the given
boundary values on xrx, will be the unique regular solution of
fudy-(- a ) dx = fffdxdy,
which takes on the same boundary values. In fact, we may
differentiate under the integral sign to find Ou/lx', and by
substituting directly into the equation, and changing the order
of integration, verify that the equation is satisfied. As is well
known, the integral in the second member vanishes when (x', y')
is a point of the contour xorxl.
Hence the unique regular solution of (41) must satisfy the
equation:
u(x'y') = f(xy') + - G(xy xy') f - au
(47) XORd
-y (a2u) dxdy,
ay ) xy
since, with our assumptions, we have
f a2udy = - (afu)dxdy.
Hence, finally, if we write
Tx'y'u = - a(x'y')u(xy') --, [aX2(xy')u(xy')],
<(x'y') = f(xy') + 2- f G(xy x'y')f(xy)dxdy,
2 l~S ~yz~l~vrd