THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 
Now let the work difference from the system y= — =x = 
that from the system 6=$= 
Bruhn 2 
pee 
2 
from the two systems immediately preceding are respectively 
dk, dE 3—a 1 19 I ) d 19 
dae z= 2 pe 2K, . 
Gyo 4 errs es 3° Jax ~? 
Hence (97) gives 
on Me es eee w)é 
A) a ak tl gM 
In the same way from w,, v2, w. and Us, V3, W; we obtain 
i ier: iH, dH, ,3-a/1, 1 Chey, 
ua, y,%) = se calles SS ee 
wa’ y'} z 
Moreover, it can be seen in a moment that the displacements due to 
y= 
- da’ dy 3 
, a-—3, 19 
=> ———— iis 
) awit’ fi 
2 
d 
dy’ 
v’x andto 0=¢=- 
are in reality the same; as also those due to 
It follows that 
2 
= oy and to 6=¢= 
eds 
alien: 
oy ip 8 Oh op 
aoe oe ie 
dy atl aw’ 0} 
Ce, 8 d 
and eG Bg ee OB, = | 
AN, Al a+l ay 2 0 
If we write U for 
and V for 
adi, dH, ee I 5 el 19 
dy Ge ait, da 
193 
x) be denoted by H,, and 
: oe - 520) by E,; then obviously the work differences 
we obtain the form which it is convenient to take as the standard for this kind of strain, 
“namely, 
with 
TRANS. ROY. SOC. EDIN., 
UG) 2) — OU 
ae la als os) 
ae = 
du’ dy’ 
a+l1 2 dz 
afi , , = Il 6 d AU 
uz,y,2)= v42 oars 2, ( 
( 2 Y; ) a+ 1 9” dy ax 
Cecile) ane EU ay 
Saas oy (peor BS 
a+1 \ dat! a dy’ 
=“ [ 
+ dy’ | : 
d (ese S @ (Se aV 
dy\dy da!) a+1 dx'\ dx’ dij 
d = SO! Ge dV 
da'\dy dz) a+1 dy\da'* dy’ 
VOL, XL. PART I, (NO. 8). 
we 
UG 
0 
o| 
30 
(98) 
