eee 
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 137 
We require a potential V, becoming infinite as 1/r at (a, y’, 2’), but with no 
| other singularity at a finite distance, and such that = 0 when z==Eh. 
dz 
Ti R= Jwe-a'yPt+y-y'), 
Bene a | Ze agenda, When 22 
/ 0 
oe 
= fed yRde, when 2<2’. 
0 
d ) 
ag a | (—kje"?-)JkKRdk, when z>2’ 
Mie " 
x 
[ Kes’) TkRdx, when z<z’. | 
Jo 
We therefore begin by finding a potential 
V, =(A cosh «z+ Bsinh xz)JqkR_ 
giving 
dV, an 
je Kee NKR Oh 2h 
OYA 
= ke a Up Ol, 2 — hi. 
We obtain 
A sinh kh+Beosh xh = e7*lt-*’) 
A sinh kh — Beosh kh = e7**+2") 
ler a cosh xz’ 
sinh kh, 
ee sinh Kz 
cosh Kh 
oes a Ge Kzcosh xz sinh xzsinh “5 aR 
sinh xh cosh Kh : 
If this could be integrated with respect to « from 0 to co, we should have a 
potential just balancing at the boundary the flow from the source. 
But V, becomes infinite as 1/ch at «=0, and the integration cannot be performed. 
We may, however, subtract from V, the (constant) potential e-"’/kh, where ¢ is an 
arbitrary positive quantity. This makes integration possible, without introducing any 
flow across the boundary. 
A solution of the problem is then 
ys 2 A ie oo ORN Kz cosh Ke n sinh «z sinh Vy : ax \ ax. 
4 sinh kh cosh kh kl 
But this form of solution, while theoretically complete, is of little value because of the 
ditiiculty of interpretation. For example, it gives no indication of what on physical 
grounds we should expect to be the chief feature of the phenomenon, namely, the 
practically two-dimensional character of the flow at a moderate distance from the 
source. . 
The transformation to which we proceed brings this out as luminously as possible. 
First, it is convenient to separate V, into its odd and even parts in «, as is easily 
done by writing cosh ch —sinh «h for e~*. 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 23 
