DR. a. B. JEFFERY ON PLANE STRESS AND 
270 
so that if hx and its second differential coefficients are finite at infinity (a = 0. ,8 = 0) 
we have there aaa = aft ft = hx and a,8 = 0. 
In the absence of body forces the stress-function satisfies V 4 x = 0. In curvilinear 
co-ordinates we have 
V 2 = Jr 
^2 ^2 \ 
S + c 
0a 2 ' 7/9 2 
and, taking hx as the dependent variable, we have in our co-ordinates 
a V\ = \ (cosh a - cos ft) {— + ? ^ 2 
^2 ^2 
O . C 
■2 sinh a ——2 sin ft ~+ cosh a 4- cos ft \ {hx). 
col c/9 
Repeating the operator, a little reduction leads us to the following transformation 
for V 4 v = 0 : 
( a 4 „ a 4 a 4 a 2 „ a 
\a a 4+2 a a 2 a/9 3 + a/9 4 2 a a 3+2 a/9 2 + 1 )^ 0 .^ 
Thus by considering hx instead of x we have a linear equation with constant 
coefficients. 
Before proceeding to the discussion of its solutions, we must investigate the method 
of determining the displacements corresponding to a given stress-function, in order 
that we may ascertain whether and under what conditions these are single-valued. 
This is particularly necessary in our case, as one of the co-ordinates, ft, is itself many- 
valued. 
Adding and subtracting the first two equations (4), and leaving the third as it 
stands, substituting for the stresses in terms of the stress-function, and for the strains 
in terms of the displacements, we obtain the following three equations:— 
+ 
A/ox 
9/910/9 
•2 (\ + /x)- 
= 0, 
(9) 
k{ h °¥« +2 » h )-b{ h % +2 * hv \ = 0 ' . (10) 
.<“> 
From the last two of these it appears that we may define a new function P such that 
0P 
0a 
h 3 ^ + 
0/9 
hv, 
0P 
dft 
= h 2 ^ + 2 fi.hu, 
COL 
( 12 ) 
(13) 
V»P = 0, 
