THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. Vout 
and (2n—1)r+7/2, and it may be shown, as in the previous case, that there is actually 
‘one root with 7 between these limits for all positive integral values of n. 
Also 
n= (2n-1)r+7/2—€; cosh E= 2m — 1/2 
e=2 log (4n—1 7)/(4n—17). 
To a first approximation 
£, = log (4n—1 2) + (2n—4)xt - C8 
In addition to the roots of (27), (28) we have of course a corresponding series in the 
second quadrant, the images of these in the axis of imaginaries. 
8. Approximate forms of the n” terms of the infinite series, when n 1s large. 
It may be useful to give in terms of approximate forms for the general terms of 
(20), (21) corresponding to the n™ roots in the first quadrant. 
i and 9. 
Ut) $e : kh = 4 log 4mm + (m+ 4)ri 
sinh kh = dekh = 4(Anr)tem+im 
cosh Kh 1 4 
_ =. : = e-(n+3)mi( — 7) 
kh(cosh 2kh—1) sinh kh-2sinh?xh — (42)? 
Z Zz 
sinh xz = $(ek*)n — d(exh) — 7, 
Ae. . mz ug miz 
= 4 { (Ana) eet) 7, = (Anz) —2he -(n+4) 7, 
Hence in ¢, the general term 
F za mi(z—h) zh tile-+h) 
_ a Fork 4 (Anz) 22 +2) — (Amar) - * e-@+D~ jh t ; 
in 4, the same as this, with the factor 7/2n7 omitted. 
In both 
oF nR iRlog 4nm 
Gy«R = ——¢@-@1D7 @ oh 
2n 
(ii) p, and 8,. 
kh = dlog 4nm + (m—4)mt 
cosh kh = 4(4nm)2e—drt 
Tn ¢, the general term 
. z-h ‘ zth , 
— mi(z — h) a wTie+h) 
Bee eB (Anz) 2h o(n-3)— (Anz) - 2h e-(n-3)-7_ ; 
Qn3 
Tn 4, the same, with the factor 1/2n7 omitted. 
Qian h Gi te aR log 4n 
kKiv = ——e \""4)/h € 2h 
N 2nR 
9. The solution for arbitrary normal traction. Questions for discussion. 
The solution of the general problem of given normal traction requires the multipli- 
cation of the functions in (20), (21) by f(z’, y’) and integration with respect to wv’, y/ 
over a finite area A. There is no difficulty in showing that these integrations can be 
performed term by term, and that the resulting series converge absolutely. 
