PLANE STRAIN IN BIPOLAR CO-ORDINATES. 279 
Identifying these with (29), we have the following relations from which to obtain 
the coefficients:— 
<piM = 2c 0 + 2B 0 sinh «! cos a, + 2K cosh 2 ^ —K 
1-2.3.02 («i) = 2cj —2 (K cosh a 1 + B IJ sinh a x ) +2 sinh a x <j>\ (a 2 ) 
2.3.4.03 (a 2 ) 2 cosh a 1 . 1.2.3.0 2 (o^) = 4 c 2 + 2K + 4 sinh <j> 2 (c^) 
r (33) 
^(ri+l)(w + 2)0 n+1 (a 1 )-2cosha 1 (n-l)(n)(n+l)0„(a 1 ) + (n-2)(^-l)(n)0„_ 1 (a 1 ) 
= 2 nc n + 2 n sinh ttl <p' n (a x ) (n = 3) . 
n(n+l) (n + 2) Vvn (a^-2 cosh aj(n-l) (n) {n+l)^ n (a 1 ) + (n-2) (n- 1) (ri)^„_i(a x ) 
= 2 nd n + 2n sinh a x \^' n (sq) (^ = 1) 
_ • • • (34) 
V' i ( a i) = 2a 0 1 
(wt 1) ^»+i (“i) - 2 cosh aj W0'„ (a,) + (n-1) («j) = 2 a n (n=l) f 
20^ ( a i) — 2 cosh a, 0' x (a x ) = —2b 1 — 2 (K sinh a l + B u cosh a,) 
30 7 3 ( a i) — 4 cosh a, 0' 2 (a,) +0'j (a x ) = — 2& 2 + B 0 , . 
r • (36) 
(n + l)0' n+ i(a 1 )-2cosha 1 n0' n (a 1 ) + (n-l)0 , n _ 1 (a ] ) = ~ 2 K (w= 3 )J 
Writing out equations (35), multiplying by e~ na ' and adding, we have 
(^+l)^» + i(ai)e _ " a, -^ / »(a 1 )e- ( " +1)Bl = 2 2 a p e~” 
p = 0 
or 
(n+l)^ , n+1 (a 1 )-^ , n (a 1 )e- ai = 2e- 2 . . . (37) 
P = o 
Now, in virtue of (30), we may write the right-hand side of this 
oo co 
— 2e na ‘ 2 a p e- ya ' = -2 2 a n+r e~ ra \ 
p = n+l r= 1 
and since 2«„ cos nfi is supposed convergent this tends to zero as n increases. Hence 
from (37) we see that the limit of (a^/V^n ( a i) as n increases is e~ a ', and hence 
the functions yfr n {a.^) are finite for all values of n and tend to zero as n increases, if 
the resultant couple acting on a = a x vanishes. 
Multiplying (37) by e” a ' and adding, we have 
71—1 g n—l 71—1 
n\ls' n (otj) e (n-1)ai = 2 2 2 a p e i2q ~ p)a ' = 2 2 2 a p e {2q ~ p)a ', 
y = 0 p-0 p = 0 q = p 
