222 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
In the first two stresses the terms of order —1 are annulled by taking 
| 3,F=0 5 ‘ é L@ 
w=(he-3%2)9,F » (5) 
Then integration of (3) from —h to h gives 
d 83 nh 
(9+ 359)F = ~ pops | 2de = ee) 
and from this with (3), 
Z 3 Oe a ek E Fi 
AC) = Qu st Bhi (f-h )[ Zaz : - 3 (7) 
Another equation is required before g, is defined, but in view of the results of (1), (ii) 
we can already infer that the values of F, Y determined by (4), (6) and (5) are the 
correct principal values, since the residual stresses nn, ns are of order zero. As for 9,, 
we note that we cannot complete its definition by annulling the residual stress nn, for 
h 
the condition | ny = 2 zdz=0 is not satisfied. In order to get the remaining equation 
for g, we must therefore solve by the method of (i) for the residual normal traction 
dy. = = so as to get the equation corresponding to (i) (5). The matter might be left 
at this stage, but it may be interesting actually to carry out the process of balancing 
the residual parts of nn and ns. By so doing we shall not only define g,, but also 
obtain a second approximation to F and wb. | 
2) 
We have to introduce into the solution F’, W’, g,/ with zF’ and oe of order 0, g,’ of - 
order 1. The equations to be satisfied are 
d dy - 
-—2 ds an = 2g.N (xz) = —2),\ . . . (8) 
2 dy = ey 
Pp ae = —2h,k ar FEE 3 . : (9) 
5 y é i s ’ o 
0 =H 195 + 5 + SgcZ(u2) , (10) 
We must first find the principal value of a in terms of defined quantities. 
Now from (5), 
y= (42 — thz)d,F, at the edge. 
Hence, by Fonrier’s Theorem, 
3) ree ee tone, A 8 Dre 
= ~ FM9E(sin 55 — grsin oy + gsin gy --- --) 
Within the plate, therefore, 
Te 
Be fh Se Ay yee oe ep a 
y= eee (Yo sin Oh ~ ait sin ont Bi Ye sin yaa .) 
