THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 157 
The following table, which may be deduced immediately from the results of 
§§ 10, 11, shows the order in h of displacements and stresses arising from ,, 9, and 
| dz, 9 respectively. 
by, Fo 
$,, 9, pz | p—a 
u,v -2,0 2 1 
Ww -3, -1 il 1 
a eon OF 1 10 
ee eel 1 0 
te 0 0 0 
It thus appears that the first part of the solution gives all the displacements to a 
second approximation, and all the stresses but %z to a first approximation. With regard 
to #, it should be observed that the solutions depending on F,, F, contribute nothing 
to it, so that, within p=a, its value comes altogether from the particular solution, 
and without p =a, its value is zero beyond the immediate vicinity of that surface. 
14. Methods and results of the special case extended to the general problem of 
arbitrary normal traction. 
One feature of the solution expressed by equations (33), (34) we have already 
found useful, especially in the important case when h is small, in such a way that Bh 
| and h/a are small fractions. We refer to the explicit separation in the solution of a 
purely local element, entirely negligible except within a certain strip of breadth com- 
parable with the thickness of the plate, from an element of a persistent or permanent 
| character, with an area of influence not affected by the indefinite diminution of h. 
Another advantage of the form of solution in (33), (34) is that the particular 
solution for the space within which the traction is applied is found in such a form that 
it can be readily expanded in powers of h, so as to give the terms of positive order in 
the infinitesimal /, as well as those of negative order which were already separated 
in (29). Thus in the particular solution, or first line of ¢ in (33), the factor 
cosh Bh sinh fz 
BXsinh 2Bh — 2B) 
can be expanded in ascending powers of (, the series converging if | 28h|<|C,|, where 
G is the complex root of sinh ¢—¢=0 with smallest modulus. Since z is of the same. 
order as h, and we are supposing 8 independent of h, it is clear that the terms of the 
| series will be of ascending order in h. 
We shall now show how the solution for the general case when the given normal 
traction is a function of x,y of unspecified form may be transformed, under certain 
restrictions, so as to yield the advantages to which we have been referring as pertaining 
to the solution (33), (34). 
The problem we suppose to be the same as that stated at the beginning of § 10, 
