282 
DR. G. B. JEFFERY ON PLANE STRESS AND 
The formula for fa (eij) given in (43), which is itself a condition of convergence, will, 
however, be replaced by 
<p\ (a,) = B 0 + 2Ke C4 sinh a 2 +2 2 b' n e na2 ,.(47) 
n = 1 
if a 2 < 0. 
From (26) we see that the coefficients A n , B n , C n , D n for 2 are determined from 
<b n (a,), fa( a 3 ), <p' n {a x ), <j> n { a 2 ), and similarly A'„, B'», C'„, D'„ are determined from 
W, ( a i(*2), ^(ai), {eh), 
The values of ^(otj), fa(a. 2 ), <j>\{ a x ), <p\(oi 2 ) will give four equations to determine the 
three constants Aj, B 1? C ]5 and the condition that they shall be consistent gives one 
relation between B u and K. The values of ^(a.,) determine the two constants 
A'], C'j, and ^(“ 2 ) are no 't otherwise determined. 
We have thus just sufficient equations to determine the coefficients in (28) with 
the exception of B 0 , K, between which we have found one relation. If a 2 >0, so that 
the region considered lies entirely on one side of the axis a = 0,’we may take K = 0. 
If on the other hand a 2 <0 the condition that the stress shall vanish at infinity, which 
is h-x^0 when a, /3->0, gives one more relation between the coefficients, so that in 
either case B 0 , K are determined. 
We may therefore adopt the following method :—Insert terms of the type (24) or 
(25) corresponding to the resultant force and couple on each boundary, and calculate 
the residual stresses over the boundaries. These will now form systems in statical 
equilibrium over each boundary, and we have Shown how to determine an appropriate 
function of the form (28). 
The problem of finding the appropriate stress-function for given tractions over the 
boundaries might have been approached by investigating the values of hx and its 
normal gradient on the boundaries, on the lines developed by MichellA The direct 
method which we have adopted is, however, in most cases simpler in our particular 
co-ordinates. 
There is an exception to this rule, namely, when a boundary is free from stress. In 
this case the boundary conditions assume a very simple form. From (6) we have 
and 
WXl 
da 
02 0 
(cosh a —cos (3) — ; (hx) — sin /3— (/iy) + cosh a {hx) = p sinh a 
dp“ o p 
the solution of which is readily found to give 
hx = p tanh a + a- (cosh a cos /3— 1) + t sin /3 .(49) 
on the boundary considered. 
* ‘Proc, London Mathematical Society,’ vol. xxi., 1900, p. 100, 
