PLANE STRAIN IN BIPOLAR CO-ORDINATES. 
277 
while over a = a 2 we have similar expansions in which a 0 , a n , b n , c 0 , c n , d n are replaced 
by a' o, a' n , b' n , c' 0 , c' n , d! n . 
If the tractions applied over the circle a = a. x are statically equivalent to forces 
X, Y at its centre, and a couple of moment L, then 
The coefficients of aa, (3(3 can readily be expanded in ^ourier series. We have, in 
fact, since a } > 0, 
and 
dx sinh a sin (3 
_ _ __ Qj ___i_ - 
8a (cosh a —cos /3) 2 
8 y _ ^ (cosh a cos (3— l) 
0 a (cosh a —cos (3) 2 
sinh a x (cosh a x —cos jd) -1 = 
oo 
— 2a 2 ne~ na ’' sin n/3, 
1 
oo 
— 2a 2 ne~ na ' cos ?i/3, 
1 
1 + 2 2 e'" 1 ' cos nj3. 
Substituting these and the expansions for aa, a (3 in the expressions for X, Y, L, and 
integrating, we have 
X = 2tt 2 n (a n — d n ) e 
I 
Y = -2x2 ]n(b n + c n )e- na \ 
1 
oo 
L = — 2-n-a cosech 2 a! 2 a n e~ na \ 
The corresponding components of the resultant of the forces applied over a = a 2 
can be obtained in a similar way. We must, however, remember that in this case 
the forces act from that side of the boundary for which a < a 2 , whereas in the case 
of the first boundary they acted from the side for which a < a x . We obtain, if a 2 > 0, 
X' = -2x2 n{a' n -d' n )e~ na *, 
1 
Y' = 2x2 n{b' n + c' n )e-’‘% 
1 
L' = 2xa cosech 2 a 2 2 a' n e,~ na '\ 
