THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 183 
From (84), (86), (89) for areal force X 
m= +EX + peel) | [X(e, y', 2) 9 *x(R)da'dy' + aX | 
aa . (98) 
3 : 1 @ 7 ae 
ae? (@ - n/a, feay| |X y’, Z)y*x(R)dx'dy | 
It may be verified that these give zero stress on z= +h, and = +e =0. 
From the formule we have given, it is of course merely a matter of the simplest 
algebra to calculate any of the stresses to whatever order of approximation may be 
required, but it may be worth while to remark here that the fundamental equations 
of equilibrium (1) may be used with great advantage in obtaining the principal results. 
If, for example, we know only the first terms of iz, ry, yy, the two first of these 
equations would give the first terms of z, zy by a simple integration with respect to z, 
and then the last equation would give the first term of %. Similarly, when the first 
two terms of iz, xy, yy are known (as above), we may find the first two terms of the 
other stresses. 
29. Transmission of force to a distance. Expansions in polar coordinates. 
We have up to this poimt been considering mainly the particular solution to 
which our general source solutions lead for any given distribution of force; or, as we 
may say, we have been investigating the effect of any given force system on that 
part of the solid to which the force is applied. But it is also of great interest to 
inquire what is the effect of this force at points of the solid remote from its region of 
application. It is obvious that we obtain a sufficient answer to this question by 
retaining only the permanent terms in the source solutions, those terms, namely, 
which are given in (65) and (83). 
For force applied only at points on a given normal to the plate, these formule 
are all that we require. They show at a glance that the distant effect depends chiefly 
on resultant forces and couples, but not entirely, since 2’ and 2” occur in the formule 
for Z force, and 2”, z* in those for X force. When the force is not confined to a line, 
but is distributed over a finite volume of the solid, the result is obtained in more 
intellicible form if before integration the function x is suitably expanded so as to 
yield a series of solutions in which accented and unaccented coordinates are explicitly 
separated. The most convenient expansion of x is in terms of polar coordinates as 
given in (e) of the introductory section. 
Suppose, then, a single force applied at the point (a1,4%,,%) or (p1,%1,%), the 
components of the force being X,, Y,, Z,, parallel to the rectangular axes, or P,, 2, , Z; 
parallel to radius vector, transverse, and axis of z. We have to find the displacements 
at (o ,»,2) where we suppose p> p,. 
For an X force, the value of W is “ with vy’ given in (83), the coefticient de- 
pending on the magnitude of X being for the moment suppressed. 
