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THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATH. 15 
As proved in (c) 
2 9 
when p<a, the integral = pane cos Mw + Be nkp cos mw(KkaG,, Kad, Ba — G,, Ka Bad n' Ba) 
2 ; , 
: p>4, i. = ant cos Ma( KAS» KAS BA — I mKa Bad» Ba) . 
Now when «p, «a are both large, 
J mkAG Kp = d ~e'K(p <i) ? 
2x,/ap 
which, with its derivatives, is very small when (p —a@)/2h is even moderately large. 
Thus, in the space without the cylinder p=a, the part of the strain given by ,, 4, is, 
when / is small, insensible except in the immediate neighbourhood of that cylinder. 
The same remark applies to the strain within the cylinder, so far as it is given by the 
parts of $, , 9, arising from the second term in the value of the surface integral. 
We naturally inquire, how do these rapidly decaying parts of the solution behave, 
and what is the order of magnitude of the corresponding displacements and stresses, 
at points actually on the surface p=a? Now, taking for example the value of ?, 
in the external region, namely, 
2 = 2 (- eos ah) = G,,Kp Cos mw («ad nkad, ba —J,,Kapad Ba) : -  (30') 
we see from § 8 (i) that when p=a, the general term has the approximate form 
z-h 
= { Bh o(nth) 
nN 
mi(z — 
h 
h) z+h mi(z-+-h) 
—1-~ Bh e~ M+) 7 } : 
| A being independent of n. Moreover, each differentiation of ¢. with respect to p 
or z will remove a factor 1/n from the general term. Hence ¢hice such differentiations, 
| but no more, are permissible, if-—h<z<h. But from (4) it is clear that none of 
: 
the displacements requires more than two, and none of the stresses more than three 
| of these differentiations for their calculation. As for 9,, the general term is of one 
order higher in n than the corresponding term in ¢,, but in compensation for this only 
two differentiations are required to find the stresses. Hence, so far as the decaying 
part of the solution is concerned, displacements and stresses at p=a@ may be found 
| by means of term by term differentiation, and subsequent substitution of a for p. 
Again, considering the order of these various quantities in h, regarded as infinitesi- 
mal, and remembering that «h and «z are of order zero in h, we see that the expression 
for , in (30’) and the corresponding expression for 9, are of order h? when p=a, 
-jand each differentiation with respect to p or z diminishes the order by one. Hence 
|the displacements at p=a are of order f and the stresses of order zero, so far as they 
arise from the decaying part of the solution ?, , 9,. 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 
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