THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 225 
With tractions of these orders, of quite general form in z, the analysis of last article 
would lead us to expect that in the balancing solution F would be of order —8, 
w of order 0, and g, of order —2. But in consequence of N being simply proportional 
to z, it will be noticed that § 45 (i) (5) gives 2g,N(«z)=0, and it follows that in the 
present case g, will be of order —1 at lowest. The displacements p’, w' derived from 
the strain defined by the functions g, will therefore be of order 0, and q/ of order 1. 
Thus we may anticipate that the first two terms of p’, and the first three terms of q’ and 
w’ will be obtained in practicable forms, 7.e. independently of series associated with the 
zeroes of sinh 2«h —2kh. 
The tractions written down above may be balanced approximately by strains F’, W’. 
We require 
-29,(E,+F) = i 
| 
=) 
-29,(F,+F) + OY = 
(iam 
9 2 é d 
$(2’ —h°)3,(F + F) + ie - | 
These are equivalent to 
$(F,+F)=0 
yy = (28 — $h*z)9,(F, + F) al) 
(3,+23,) (+ F)=0 
which determine F’, W’. 
The principal terms of the residual tractions are 
i ep — 2h*B(z/h) = F(F, +P) | 
as 
co 9 dy’ (2) 
Ba SE pana +F) | 
as in (ii1) (12) of last article, and they may be dealt with in the manner illustrated 
there. 
To balance them take strains defined by F’, W’, g,’ with zF’, ‘ = i ,9, of order —1. 
We must therefore have 
n° B(olh)T9,(Fy+F) = —29F" +3y eae 
(212 |p)B(e/h)9,(Fy+F) = -29,0" + PY (3) 
o= 1 nae 429 /Z(«2) 
Z ds dz j 
From these, as in last article, 
por 38d 
3,F —— qe Vsltz Fol Fo a5 18) | 
Wee 384 df 1 
(93+ 9)F ae wa ~— 3,(Fy + F’) \ (4) 
y =(52- = are 2), Won? h92(y +F \(sin 5 = - gains + vee | 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART ‘ (NO. 8). 34 
