THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 211 
We see also that the principal part of the displacement is of one order higher 
in / than the principal part of the traction. 
(ii) Dilatational transitory modes. 
There would be some advantage in working with the functional symbols 9, ¢, 
as with yin the last case, but on the whole it seems clearer to deal with a typical 
solution corresponding to a single root « of sinh 2«h + 2«h, 
b=cosh xzf(a, y) ; 0=cosh 2xh- 
sa 
4 (cosh 2«h + a) cosh xz + 2«z sinh xz | 
w= ei (cosh 2«h — a) sinh xz + 2«z cosh xz } | 
ae - “| (cosh 2«h + 3) cosh Kz + 2«z sinh KZ } 
oa :{ (cosh 2«h + a) cosh xz + 2«z sinh Kz } 
dy” 
“_ D} . 
Dy = = a { (cosh 2kh + a) cosh «z+ 2«z sinh «z } 
oe ae { (1 + cosh 2«h) sinh xz + 2«z cosh Kz } 
Hence 
a = “| (cosh 2«h + 3) cosh Kz-+ 2«z sinh xz } 
Way ey =) { 2 49 Fall 
-(— in? ds Is (cosh 2«h + a) cosh «z+ 2«z sinh xz J 
ns _ ddf_ 1 — 9 ae 2 
Du - ae aide (cosh 2«h + a) cosh «z+ 2«z sinh xz | 
me ee | (1 + cosh 2«h) sinh «z+ 2«z cosh kz \ 
Thus at an edge where the rate of variation of f is of the same order in h as f 
itself, say order zero, the normal and perpendicular displacements are of order —1, 
while the tangential displacement is of an order one higher; the normal and 
perpendicular tractions are of order —2, the tangential traction being of order —1, 
or again one higher. 
Hence this type of strain contributes most to those components of displacement 
and traction to which the ¥ type contributes least, at an edge. 
(iv) It is now possible to assign approximately to each of the three types of strain 
the portion which it carries of any given distribution of edge traction. Let this 
distribution be N,S, Z, functions of z,s, of order zero in h. We can satisfy the con- 
ditions to the first order by a solution in which the principal part of the traction 
