144 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
(x, y, 0) and the limiting value of the normal attraction at (#, y, «) as « approaches 
ZeX0. ) 
As a preliminary to the general problem, we take therefore the special case in which 
z2 on 2=h is equal to ¢e/{(x—a’)?+(y—y')’+e"}', or, in the form of a definite integral, 
Jee "KJ (KR) dk, 
where 
R2=(@-—2')?+(y-y)* 
Making a further reduction, we begin by taking, in place of this integral, simply the 
function «JR. 
The function ) is not required, and ¢, @ are of the forms 
= (C, sinh «z+ C, cosh xz)JkR 
6 =(C, sinh «z+ C, cosh xz).Jy«R } 
In accordance with (5) these satisfy the conditions 
dd dob, 2S 0 on z= +h | 
Be) 
a ae 
2 2 3 ; al i 12 
oe -S + 2 $0 on z= —-h | (1 
Ze 2 
=«J9kR/2 on z=h) 
Hence we easily find 
; Aken ae cosh Kh, J es s 
Be «(sinh 2xh — 2h) oe as 
sinh xh, : P 
«(sinh 2«h + 2h) Sucieore «| (13) J 
sosh Kh + 2«h sinh Kh z 
Aghia. SSE i eee h xz 
f «(sinh 2«h — 2«h) NS ae 
sinh kh + 2«Kh cosh hy 
R cosl | 
«(sinh 2«h + 2«h) ote cosh KZ 
If these expressions, multiplied by e~“, could be integrated with respect to « from 
0 to o, we should have at once a solution of the preliminary problem. But this 
integration is not possible, owing to the nature of the functions of « near the lower 
limit «<=0. In fact, if the values of 4u@p, 4u6 in (18) be expanded in ascending powers 
of x, the expansions will contain terms in 1/«? and 1/k, so that near «= 0 
4d = H/x? + K/x + terms of positive degree 
4.0 = L/K? + M/«+ s Ap 
These terms of negative degree are potentials contributing nothing to the stresses on 
z= +h, as we see from (12), since «J,«R contains no terms of negative degree. They 
might therefore be subtracted from the expressions (18) without affecting the satisfac- 
tion of the conditions in (12). This simple subtraction would, however, introduce 
terms not integrable right up to the upper limit, at least after « is put equal to zero, 
as eventually it will be. The difficulty is met by subtracting from 4uq@, not H/«*+K/k, 
but H/k+Ke-/«; and from 440, not L/k?+M/«, but L/e + Me-/«. 
