THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 163 
results at once. These are, if we separate the parts odd and even in z, 
sinh xz 
- 2- 2) ayers «A sinh kh Ces sear \ ; ’ i ae) 
where « is a positive imaginary root of cosh kh, 
pz CR BS ie ie xe, ees 
(—) 2h (cosh Bch — 1)00%® — re BeGK' SS x) ~ 40°V X . (49) 
2«h cosh kh — sinh xh)sinh xz 3 1 it 
-2 «h(cosh 2xh — 1) MS Ah? sya %x "EG OMN 2129? ) a qo°V x PEK 
where « is a zero of sinh 2«h — 2«h, with positive imaginary part. 
cosh Kz 1 I es h 
= 2(- 2) «2h cosh Zh cosh ch0\*B) fc i(x Ba \- 6VX ; i - (50) 
where « is a positive imaginary root of sinh kh. 
; cosh xh cosh xz 1 ee ales ie 
Pia 2 aE ?h(cosh 2xh + 1) Rgds Ta(x - a9) 2x) + o4V x (51) 
Ae > Cet sinh xh — cosh kh) cosh xz. R 1 11h : ; 
te on kh(cosh 2xh + 1) Eley ax - 5 Seu *x)- 94 VX 
where « is a zero of sinh 2«h + 2«h , with positive imaginary part. 
The solution is defined by these equations with 
17. Composition of the solution. 
On examining the composition of the solution, we observe in the decaying parts of 
?, 9, solutions of the class already obtained in (22), (25), and in the corresponding part 
of Y , solutions of the type 
y = sin(2n+1) Se 8 y) 
Nre 
y = cos" F(x, y), 
each giving = =0 onz=+h, and therefore zero tractions at the surface. As for the 
permanent terms, they may be arranged in the following groups, in each of which the 
surface stresses vanish. 
: oa 1 
i = - 50x) 
( ) 2 Te ax 6. vex 
= +5, aC - Bux 2ey%y) 
* 1 d 
ll Spee 
m) — %> “90% Gn 
ee ORs 
es 40 dx” x 
