172 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
The particular solution is 
{ + ip cosh B(z - 2’) 
cea) = Qa m pP cos Ma arn Bz : b 
| : BX(sinh 28h — 2 pny? sinh Bz -4a+ cosh 2Bh cosh Bz’) | 
cosh Bz (70) 
B%X(sinh 2Bh +2 any 2 cosh 82’ +} cosh 28h — a sinh Bz’) 
with a corresponding expression for 9, obtainable from (61) by changing « into 6 and 
1 
then replacing JbR by B 27) ,8p COS Me . 
It is easy to verify that this is actually a particular solution. Consider in the first 
place the analogous forms of ¢, @ in (61), and for greater generality, suppose Jo«R 
replaced by f(x, y) where (V°+«") f=0. 
Then, from the method by which (61) were found, it is obvious that they give no 
stress across the planes ===. Let us examine the effect of the discontinuity in the — 
forms of ¢ , 6 at the plane z=’, on the displacements and stresses as given in (4), (5). 
If we take simply 
1 
p = 9, cosh x(z- 2) f 
6 = ae cosh k(z—2)f—2 sinh k(z-2')f 
then we find at z=2’, 
“ney =m = 0 
me = wy = 0 
ae = — plat )ef. 
Thus with the complete expression (61), the displacements are continuous, as also the 
stresses iz, %y, but x (z=z' +) exceeds xz (z=z’—) by —2u(a+1)kf. We thus see that 
in (70) the corresponding discontinuity in % will be —47x(a+1)J,,8p cosmo. This 
continuity of displacement, and discontinuity in 2, are precisely as demanded by the 
conditions of equilibrium of the plate. : 
If we take (61) with Jy«R unaltered, prepare them for integration as in § 19, multiply — 
by e-“ and integrate with respect to « from 0 to «, the discontinuity in z at z=2' will — 
become 
— Qua + vf, e-*eKJ kKRdk . 
If further we multiply this by Z(a’, y’, 2’)da’ dy’ and integrate with respect to a’, y’, © 
and then take the limit for «= 0, the discontinuity becomes, in virtue of (11), 
— Arp(a+1)Z(a, y, 2’). 
We have thus a proof of the solution for an areal distribution of Z force, independent of 
the infinite solid solution (6), which might itself be found from the beginning by this 
method. 
