194 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
(ii) Transitory modes, ¥ or rotational type. 
Referring to the expression for /’ in (82), put 
E, = work difference from the system y= = 
aah cosh xzGoxR. 
Then for this part of the solution 
u(a, y’, 2) = 2 Th cosh x 
ne, x, 2) = 2(- 198 oo 
w(a’, y', 7) = 0 
d Ey tae E 
ay? 
The solutions here are obviously of the at is = cosh «2 His (2’, a, 
where « is a pos. imag. root of sinh ch, and 733+ 7 a +E, =0. 
(iii) Transitory modes, 6-¢ or dilatational type. 
Looking to (64), (82), put 
E, = work difference from the system a= cosh KzGykR 
: Sarpu(a+ 1)«?h(cosh Qkh + 1) 
@ = cosh 2kh -f 
Then 
fg GR Sau, 
u(x’, y', 2) =, Des 
bpapeirie Be dk 
u(x, Ys #) = a 
2x2’ sinh x2’ + (cosh 2«h + a) cosh xz’ 
Ue, 52) = DuKB yf 2! cosh xz’ + (cosh 2«h — a) sinh xz’} 
Ae PE, @E, 
where « is a zero of sinh 2h + 2«h with pos. imag. part, and 7 o+— 3 iy +E,=0. 
The solutions are of the type ¢ = cosh xz, (a’, y’), @=cosh 2h. 
II. Flexural part of the solution. 
(i) Permanent mode. 
Let F, = work difference from the system 3 
== es 1 3~ 2 
- a a Tie x) 
3 ge : 
7 B2auhe s(x- po ae *) 
Then VF, = work difference from ¢ = — sa? Vx=—-8, 
and Tepe, f , aF, 2 at+5 he aa r 
ue, ys 2)= — 2 Ge + aay (Sy et Sy Wd yap VE, 
dF, a, (arog, | O41) 
ve’, y', Z)= -2 
at aeik iD 40 Wea VE 
w(x, y’, 7)= F, + {4n?- 
J 2 iD I 
2 ai (2? —h2)}V oF, 
