THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 197 
| fact that in the solution for a single force in any direction, the component displacement 
in that direction is symmetrical in the accented and unaccented coordinates, a theorem 
analogous to a well-known property of Green’s function in Potential Theory. 
It is interesting to observe that, in the process suggested in the last sentence but 
_ one, we only need to know the comparatively simple source solution for a single Z 
_ force in order to deduce the w displacement for any system of forces and face tractions 
_ whatever. 
36. Application of Betts Theorem to the problem of given edge tractions. 
In the remaining pages, we shall be occupied almost exclusively with deformations 
of a finite plate under edge tractions only. For brevity we may refer to such deforma- 
| tions as free. 
The formule (97) express the internal displacements in terms of the edge dis- 
| placements and edge tractions. We may indicate here the general lines along which 
we naturally proceed in the attempt to reduce these formulz to expressions in terms of 
| displacements alone or of tractions alone. 
Taking the first equation of (97), for example, if we wish a formula containing edge 
displacements only, we look for free displacements in the form of functions w,’, v,’, wy’ 
| of x, y,z, such that u,+um’, 1, +0,', w,+w,’ shall be equal to zero at the edge. 
If X,’, Y,’, Z,’ be the edge tractions in the system w,’, 0)’, w;’, then by Betti’s Theorem 
| | (Xu + Yo’ + Zw) -Xju-Yy'v- Z;w)d8=0, 
, and by addition of this equation to (97), 
u(e', Ys d)= = | { | WX +X))-4 (y+ Vy) + 0(Z, +B) bas. 
The problem of arbitrary edge displacements is thus reduced to a problem in which 
these displacements have a comparatively simple form. 
When we attempt to find a formula in terms of edge tractions only, the procedure is 
not quite so simple, in consequence of the fact that the tractions X,, Y, , Z, are not equili- 
brating, but equivalent to a negative unit X force through (#’, y’, 2’). From various 
methods of meeting this dithceulty we select the following as the most convenient in the 
| present case. 
We have seen in § 30 that the system %, , v, , w, can be decomposed into four systems. 
The first system, say U,, V,, W, conveys no resultant stress; the second system conveys 
|a stress equivalent to a unit X force through the origin, and the displacements are 
independent of x’, y’, 2’; the third system conveys a couple z’ in the plane zOx, the 
displacements contain z’ as a factor, but are otherwise independent of 2’, 7’, 2’; the 
fourth system conveys a couple —y’ in the plane wOy, the displacements involving 
x’, y',2 only in the form of the factor y/’. 
