THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 201 
The first series of residues is the function v of (iv). Thus we obtain, (>) 
4 cs, cosh «(2h —€) cosh KG, 1 ' ; 
~ rp x sinh 2«h a3 (Joa)? Jo pido xp (3) 
vi 
9 
Spe Peale PP1 2 ON RNS ut DN ay 9 
=e pipe + seh (a? + 4h?) or, aie = spe) aa (py? — 4p, 4;") 
When (>, we have merely to ee ¢ and G in this formula. 
We may verify in a moment from this, as from the perfectly equivalent form (iv), 
that the internal couple is balanced by the stress at the cylindrical boundary, and that 
there is no stress across the plane ends. But if we remove the last three terms from 
(vi), we make no change in the internal singularity, these terms being the same whether 
¢ or G be the greater. 
We thus obtain the displacement when the internal couple is balanced at the plane 
ends, namely, (>) 
ee cosh «(2h - £) cosh x, 1 \ 
EEE di : k 
Tad ~ sinh 2«h Ka3(J Ka)? Jospin? (vii) 
v= — Be pypg + 
Ta 
Here, as in (vi), the summation extends over the positive roots of 2J,/«a +KaJ Ka. ‘The 
solution for symmetric transverse traction ,, , 2, on the ends, which might be obtained 
in an abnormal form from (iv) with the cognate formula for p> p,, is given in normal 
form by a direct application of Betti’s Theorem to (vii). 
Thus 
cosh 1 a : 
v(p, ,4)=— ao Sent aI, «ay? Ih Qn;,pJ 9 Kpdp 
a cosh «(2h— €,)Jq'kp, 1 ip o 
Ya ~ a a aT «ay? QopTy Kpadp (viii) 
- a pio in Qop*dp 
The result belongs rather to the theory of a long rod than to that of a thin plate. The 
permanent term depends only on the integral couple, and coincides with that given by 
Saint Venant’s theory of Torsion.* 
38. Problem 2. Boundary values of the normal displacement u, and the shearing 
stress normal to the plate x, are given functions symmetrical about the amis ; 
the displacement v, or the shearing stress po, vanishes. 
We begin with the case of a solid cylinder. 
(i) Permanent extensional mode. 
Referring to § 33, I. @), we see that under the conditions proposed the function 
E, must vanish, and the solution in cylindries is 
duo), 
Chey - HP) 10m 
dps ee, ”) Vi 
U(e,, 2%) = 
a-—3 5 
(Py > %) = ane +E, 
* The writer hopes to publish shortly a solution of the problem of equilibrium of an infinite circular cylinder, 
in which the celebrated solutions of Sarnt Venant will appear as the leading terms. It will be shown that in a 
finite cylinder the permanent modes are given exactly by Satnt VENANT’s theory. In the theory of thin plates, the 
permanent modes can only, in general, be found approximately. 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 31 
