212 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
is of order zero for each type of strain. For, taking account only of these principal 
parts, the equations to be satisfied at the edge are on this supposition 
N=N, +N, 
Ss =5S,+S, 
L= La 
the suffixes referring to the permanent, rotational, and dilatational types respectively. 
Now 
h 
dy 
dz | - 
dy h 
Die Z = 2 =0. 
8,= Peas and | e S,dz= 2p a 0 
Also 
N,z= = 2a ae { (cosh 2xh + 3) cosh «z+ 2«z sinh xz) i 
he 
i N,dz=0 
-h 
1 fica 
No-a] Ndz 
1 h 
Ss = Ih ie Sdz 
and, as mm art. 40) 
Hence the above equations give 
and these conditions determine the permanent mode. 
~ can now be found from the boundary condition 
a ay =| ‘ 
2p 73 =S- alee Sdz. 
For, taking 
NTre 
2 ; pene 
the condition is 
h? i fe ; 
2 tn" n? cos = Bones 18-5; _ ole , 
Now the right-hand member here is a function of s, z, even in z, the z-integral 
of which from—/h to h is zero for all values of s. It can therefore be expanded 
by Fourier’s Theorem in the form, valid from z= —h to z=h, 
VY, is then determined as satisfying Tint oy, =0 throughout the plate, and taking 
the value A,/n’ at the boundary. 
Lastly, the equations to determine the dilatational mode are (since at the edge 
of -ixf to the first order), 
i 
2p 
oa ui ike \ (1+ cosh 2«h) sinh «z+ 2«z cosh xz} == ly 
2p 
J eee 
zy ve 4 (cosh 2«h +3) cosh «z+ 2«z sinh xz } =- | n-1[" naz t 
