THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 167 
Assuming 
¢=A sinh «z+ B cosh xz } TR, 
6=C sinh xz+ D cosh xz § ee 
(57) give four equations to determine A, B, C, D. By addition and subtraction 
these are resolved into two equations for A, C, and other two for B, D. Thus we 
find 
_ sinh xzJ9kR 
«(sinh 2«h — 2h) 
| xz sinh xz’ - 3 cosh x2’ (e~ 2x + a + 2«h) \ 
_ cosh KzJgkR 
«(sinh 2«h + 2«h) 
— sinh Jo R - 
~ «(sinh 2«h — 2«h) 
xz cosh xz’ +4 sinh xz’(e~?*" — a — 2h) 
( 
j 
| — «sinh xz'(e- 2h +. 2«h) + cosh wa So + “ +axkh+ arch) ; 
Seah ah 3h) { Kz’ cosh xz’ (e~ 2k? — I«h) + sinh Ke - a3 SEEN =, takh + 2e7h? ) t 
(58) 
If these expressions could be integrated with respect to « from 0 to «, the balancing 
displacements U, V, W of (55) would be determined. Near the upper limit the 
functions converge to zero exponentialwise, since both z and z’ lie between —h and +h. 
But for «=0 both functions are infinite, and their expansions in ascending powers 
of « contain terms of negative degree which must be removed after the manner of § 3. 
The integrals are then convergent, but a further modification of a different sort is 
necessary before they can be transformed into series as in S§ 5, 16. The possibility 
of this transformation in the former cases was intimately related to the fact that the 
functions in (13), (46) were odd in «, which the functions in (58) obviously are not. 
However, when the odd and even parts are separated, the latter are found to have 
a very simple form, free from the denominators sinh 2«h — 2«h, for we find 
b= sinh x(z—2’)TyeR 
Bn 
sinh «K2zJ «R : , 
Sa oh “gt = s sk (4 
«(sinh 2«h — 2h) { east 2 Kee cosiaszxdd)) cost \ 
cosh Kad KR 
«(sinh 2xh + 2h) 
{ kz’ cosh xz’ + 4 (cosh 2«h — a) sinh xz’ \ 
=— \ = sinh «(z— 2’) +2 cosh x(z— 2’) \ JokR 
K 
sinh xzJy«R 
«(sinh 2xh — 2xh) 
cosh KzJ okR 
x(sinh 2«h + 2h) 
{ — «xz cosh 2«h sinh xz’ + ie cosh 2«h + a + 2xél* cosh Kz \ 
a 
“= 
| kz’ cosh 2«h cosh xz’ + ( = = cosh 2«xh + ea + 2c? Jeosh Kz. \ 
(59) 
The even terms in « can be eliminated from these expressions by including the 
