THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 
(ii) Rotational transitory modes. 
These are as in § 42 (ii), except that now 
ie Te 
y= PACE y) sin re 
where Vb, — (2n +1)? els 
(iii) Dilatational transitory modes. 
¢=sinh Kz f (#5 Y) 3 6= —cosh 2kh- 
As in § 42 (iii), 
(a — cosh 2«h) sinh «z+ 2«z cosh =| 
w = «f { —(a+cosh 2«h) cosh «z+ 2«z sinh xz} 
5 = ~Kf{(3—cosh 2«h) sinh «z+ 2«z cosh «z} 
2m 
-(“ GE ae uy ) \ (a — cosh 2h) sinh xz + 2«z cosh Kz ; 
pin ds 
a = EZ . a ; “| | (a — cosh 2h) sinh «z+ 2«z cosh Kz t 
es = ue | (1 — cosh 2«h) cosh xz + 2«z sinh Kz ; 
Qe “dn 
df ; 
We have i= 7 tf to the lowest order, 
+ of 
Be a di. 
: df 
Hence if we put = Ie 
2{(cosh 2kh — 3) sinh xz — 2«z cosh xz} = N(kz) 
(1 — cosh 2«h) cosh xz + 2«z sinh Kz = Z(kz) 
the above strain gives 
~ 
——= JN (kz) : 
2 gene with an error of relative order h , 
ns = 0 
3p = 94(kz) exactly, 
and we may note that | i aN (xz) =0 and | ‘ Z(kz)dz=0. (§ 41.) 
- —h 
7A 
The same remarks as in the extensional case might be made here about the 
complementary character of the types (ii), (ii) in regard to their contributions to 
edge displacement or traction, when / is small. 
(iv) If we follow the lines of the discussion of the extensional case, we have now 
to consider the approximate allocation of a given system of edge traction among the 
three types of strain. 
