THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 215 
tractions required to maintain the permanent solution alone?” or the practically 
equivalent question, “ What conditions must the edge stress satisfy in order that the 
permanent mode may be absent from the resulting strain ?” 
The expression found for U’ may be transformed so as to supply answers to these 
questions. 
In the first place, we note that the values of p’, q’, w’ will still be correct to the 
order stated (but will contain superfluous terms) if in the expressions just given for 
them we write 
Wie WSS 10) io ees nle 
and 
Aga Ay BAG ton yAy 4A; s 
Write also 
for WF +i, A for Ay+A,+A,, 
and similarly for the other quantities. 
Hence 
,d 
p —=P— 402" as 
the error in each case 
qg=Q+ hoo being of order h? 
w= —ozA 
and 
4 U , “rt ot 
U'(e', y',2 =| {{ x(P- po2) +8(Q+ hor == Boils | ds de < : (4) 
d P dP : ; 
_ Also from (1 gsi —~, -N+—- 2 are of order h, and the formula for U’ will there- 
ds p ds p 
fore be correct to the same order as before if we substitute at for A, and 
ans for Il, We shall also write 
Thus 
Oe, y,2)\= fas) xp - jon, (= - 2) +8,Q+ bo8, gant -)~ (Ges) . ©) 
dN, ds, 
TFN, , 8,,2,, 72) = 
this into 
da { P(N, 72,40 Ne _y 7 aS, ioe Sh 
! : ( f pee ac Ss hip ds ) +@(8, Wid ds 456 ds pee 7a) ° ; (6) 
are continuous over each edge line, integration by parts transforms 
