THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 207 
There are other expansions which we know must exist, but the coefficients of which 
we cannot determine. The following examples are of special importance, and will be 
found useful immediately. They refer to the case of a plate bounded by a single 
infinite edge, filling, say, the region «> 0, and the displacements considered are such 
that u, w are functions of x, z only, while v vanishes. 
Extensional modes. 
6 a—3 
The permanent mode is of the form w=2,w="—4:. 
In a transitory mode with ¢ =e" cosh xz, 9= cosh 2xh- p, we have 
Baie =e “(cosh 2xh + 3 cosh «x + 2«z sinh Kz) 
~ 
we 
Qpine— = e'**(cosh 2kh + 1 sinh xz + 2«z cosh xz) 
As a special case of the results of last article, it follows that the coefficient of the 
permanent mode is determined from the given value of iC wedz at the edge; and 
this integral normal stress is zero for each of the transitory modes. 
Hence if P(z) be any even function of z, with i SS P()dz=0, and Z(z) be any 
odd function of z, coefficients C, exist such that at the same time 
; Zaps C,ie**(cosh 2xh + 3 cosh xz + 2«z sinh Kz) = P(z) | 
and ea uw > C,e**(cosh 2h £1 sinh xz + 2«z cosh xz) = Z(z) | 
Flexural modes. 
In the permanent mode F of equation (95) is of the form Aw?+Ba*. <A and 
B are found from the edge values of ie 2. eedz and i ede. These integrals 
—h 
vanish for a transitory mode, in which, with =e sinh «z,@= —cosh 2«h- p, 
= 
CR 
Dah e ie**(3 — cosh 2«h sinh xz + 2«z cosh x2) 
UZ 
Tne e*@(1 — cosh 2«h cosh «z+ 2«z sinh xz). 
We infer that if P(z) be any odd function of z, with | * #P@) dz=0, and Z(z) any 
. . h . . 
even function of z, with i _ Aida=0, values of C, exist such that simultaneously 
aa > Cde'%(3 — cosh 2«h sinh xz + 2«z cosh Kz) = P(2) | 
a > C,e%**(1 — cosh 2kh cosh «xz + 2«z sinh xz) = Z(z) | 
(ii) 
The limit for «=0 may be taken term by term, provided the resulting series 
converge. In the following analysis we shall assume that they do so, but this is 
merely in order to avoid lengthy forms of statement; the argument could be put, if 
necessary, in a form independent of this assumption. 
