276 
DR. G. B. JEFFERY ON PLANE STRESS AND 
corresponding to forces and couple applied at a = — c°, and similar terms in 0 2 , log r 2 
corresponding to forces and couple applied at a — + go. The corresponding forms of 
/?X can be expanded in series which are included in our general form, but here again 
the expansions are different on opposite sides of a = 0 and diverge for a = 0, j3 = 0 
together, i.e ., at infinity. This divergence corresponds to the obvious fact that forces 
or couples must be applied at infinity to maintain equilibrium. 
Owing to difficulties of this kind we shall find it convenient to insert the 
appropriate terms corresponding to the resultant force and couple over a boundary 
and to investigate the stress-function corresponding to the remaining applied forces 
which will be in statical equilibrium for each boundary. 
Let us write for brevity 
<p n (a) = A n cosh (n+ 1) a + B„ cosh [n — 1)«4 C„sinh (n+ l) a + D„ sinh {n— l) a 
(a) — A' n cosh (n + 1) a + B'„ cosh (n — 1) a + C'„ sinh (n +1) a + D/ sinh (n— 1) a, 
if n ^ 2 and 
0! (a) = Ai cosh 2a + Bj + C x sinh 2a 
xlsi (a) = A\ cosh 2a + C\ sinh 2a. 
Setting aside the terms corresponding to the resultant forces and couples over the 
separate boundaries we have 
hx = {B,,a + K log (cosh a —cos /3)} (cosh a —cos (3) 
oo 
+ 2 {(p n (a) cos n/3 + i/r n (a) sin n(3] . (28) 
n = 1 
where the term in K may be omitted when the region considered lies entirely on one 
side of the line 
§ 4. Boundary Conditions. 
Let us consider a plate bounded by two curves a = a 1} a 2 . We may suppose 
a 1 > a 2 and a. x > 0. Then, if a 2 > 0 we have a finite plate bounded internally and 
externally by circles which are not concentric, if a, < 0 we have an infinite plate 
containing two circular holes, and by suitably choosing the values of a, a 1} a 2 we can 
make the circular boundaries in either case of any desired radius and centre distance. 
In particular if a 2 = 0, we have a semi-infinite plate bounded by a straight edge 
and containing a circular hole. Suppose that such a plate is in equilibrium under 
given normal and tangential forces applied over the boundaries a = a l5 a 2 , so that we 
are given over a = a u 
cia/3 = a 0 + 2 ( a n cos n/3 + b n sin n/3), 
1 
^ v 00 / 
«aa = c 0 + 2 ( c n cos 71/3 + d n sin n/3), 
( 29 ) 
