278 
DR. G. B. JEFFERY ON PLANE STRESS AND 
If a 2 < 0 there are some differences of sign owing to the different Fourier expansions 
for the direction cosines. We have 
X' = — 2-7T 2 n ( a' n + d ' n ) e nCLi , 
1 
Y' = — 27 r 2 n (b' n —c' n ) e na * ! , 
1 
oo 
L' = — ( 2j'kcl cosech 2 a 2 2 a'„e n02 . 
1 
Hence, if the forces acting on each boundary are statically in equilibrium, we have 
with, if a 2 > 0, 
or, if a 2 < 0, 
2 n (a n —d n ) e~ na ' = 0, 
2 a n e~ na ' = 0, 
1 
2 n(a' n -d' n )e- na * = 0, 
1 
in{a' n + d' n ) e n ^ = 0, 
2 n{b n + c n )e~ nai = 0, 
1 
2 a' n e ±n “ 2 = 0. 
2 n {b' n + c' n ) e na "- = 0, 
2 n (b' v —c' n ) e 1ia '- = 0. . 
(30) 
(31) 
(32) 
We will now show that it is possible to determine a stress-function of the form (28) 
which gives the appropriate stresses over a = a 15 a 2 , and which gives no stress at 
infinity if the region considered extends so far. 
By the aid of (6) we can calculate the stresses corresponding to the stress-function 
(28). We obtain 
2 aaa — K (l — 2 cosh 2 a) — 2B 0 sinh a cosh a. + 2 <p l (a) 
+ 2 (K cosh a + B 0 sinh a) cos /3— K cos 2/3 
[(w+ 1) (n + 2) 0 n+1 (a) —2 cosh a (n 2 — l) <p n (a) 
+ (n— l) (n — 2) (a)] cos n6 
+ 2 <J + [(n+ l) (n + 2) i/r n+1 (a) — 2 cosh a (n 2 — l) \p- n (a) 
+ (n— 1) (n— 2 ) V^n-i («)] sin n/3 
— 2 sinh a [^>' n (a) cos n(3 + i/r'„ (a) sin n/3]. 
n = 1 
and 
2«a/3 = i/r'j (a) — 2 (K sinh a + B 0 cosh a) sin (3 + B 0 sin 2/3 
® f [(w+l) V^'n +1 (a) —2 cosh a.7l\Js' n (a) + (n— 1)^-1 ( a )] cos 
71 = 1 ^ ~ [( n + l) ^n+i ( a ) "2 cosh a.n<p'„ (a) + (n— l) </>'„_! (a)] sin nf3. 
