158 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
but with f(x, y) instead of J,,8 cos mw», and with any continuous area A instead of the 
circle within p=a. 
The solution will then be of the form defined in (29), (30), but the integral of (29) 
will now be 
B= [ [xR y/)ae'ay’ 
and the integral of (30) 
me i i Gy(«R) F(a", y/)da’dy. 
Since y2y(R)=log (R/2h) we have 
VIF = V7(VF) =2af (2, y) 
also (y?+x)I = - Inf (zx, ) \ 5 : s : 9 (35) 
If f(x,y) and its derivatives of the first two orders are finite within A, we may 
transform I by Green’s Theorem. ‘Thus, excluding from the area A an infinitesimal 
. ao § eh Genes 
2 
circle about (# , y) as centre, and writing V° for 7 + dy? 
eee 5 | i Hal, y' 7G xRda'dy 
or 
ree =| [Gockw Vt, ded dy’ 
= 2 | ' Ce 2G - GyeRS Fe’ y \ ds ma: - (36) 
25,9 
the line integral being taken round the boundary of A. 
If this three-termed equivalent of the integral I be substituted in the series for 
, and @,, each of these series may be subdivided into three, $, for instance into 
j 1 cosh «ch sinh xz ; rei! 
| Crk Wo SAE OF Yoel ; 
(i) mK? Klu(cosh 2xkh — abl IKK VY A(x’, y')du dy, | 
a series of the same general form as the original series, but at once more convergent, 
and of two orders higher in h ; 
5c 1 cosh «A sinh xz aim Ch d 7 
(ii) = 2 Kh(cosh 2xh — 1) ve ae ane s ler As y) et 
which corresponds to a strain local to the boundary of A ; 
are 1 cosh xh sinh xz : : : 
(iii) afl, you @ah(eosh2ah—1)? ® Series which can be summed in the same way 
as (31), being in fact simply the first series of (31) with 8=0. The sum is therefore 
— cosh xh sinh xz 
D) ’ fficient of x° i WL OS Te ON 
mf(x,y) » coefficient of x” in «(sinh 2xh — 2xh) 
We may now, by repetition of the same transformation, obtain a similar threefol 
