PLANE STRAIN IN BIPOLAR CO-ORDINATES. 
283 
The relations (48) and (49) are the necessary and sufficient conditions that a 
boundary a = constant should be free from stress. The constants p , <r, r are 
Michell’s three constants of the boundary. 
§ 5. A Cylinder or Pipe with Eccentric Bore. 
In this section we will consider the problem of a cylinder, whose cross-section is 
bounded by two non-concentric circles, which is subject to a uniform normal pressure 
over its internal surface and a different uniform normal pressure over its external 
surface. By Filon’s theorem of generalised plane stress precisely the same analysis 
will give the average stresses in a plate of the same section under the same applied 
forces. 
Let the boundaries of the cross-section be defined by a. — a. x for the internal 
boundary and a = a 2 for the external boundary. Then a 1? a 3 are positive and a,>a 2 . 
Let the applied pressures be Pj, P 2 respectively, so that aa = — P x on a = a u aa = — P 2 
on a = a 2 and a/3 = 0 on both boundaries. 
Let us assume 
= B 0 a (cosh a — cos /3) + (A x cosh 2a + B : + C x sinh 2a) cos /3. 
Calculating aa, /3/3, by means of (6) and applying the boundary conditions, we find 
the following values for the constants :— 
B 0 = 2 a M (Pj —P 2 ) cosh (a x — a 2 ) 
Aj = -aM (Pj — P 2 ) sinh (a^ao) 
Cj = aM (P x — P 2 ) cosh (a! + a 2 ) 
Bj = aM {P x cosh (a, —a 2 ) sinh2a 2 — P 2 cosh (a] — a 2 ) sinh 2a 1 + (P 1 + P 2 ) sinh (a, —a s )} 
where, for brevity, we have written 
M = \ cosech (a. l — a 2 ) {sinlffaj + sinlffa 2 } -1 
The most important aspect of the problem is the value of the stress /3/3 in the boundaries, 
for it is upon this that the strength of the cylinder will depend. This is most readily 
determined by (7), and we find without difficulty 
aa— /3/3 = 4M(Pj — P 2 ) (cosh a —cos /3) {sinh (a, + a 2 —2a) cos /3 — sinh a cosh (a,-a 2 )} 
so that on a = 
/3/3 = — Pi + 4 (P x — P 2 ) M (cosh aj— cos /3){sinh (a x — a 2 ) cos /3+ sinh a Y cosh (a, — a 2 )} (50) 
and on a = a 2 
/3/3 =— P 2 —4 (P x —P 2 ) M (cosh a,—cos /3){sinh (aj — a 2 ) cos 8 — sinh a 2 cosh — (51) 
2 R 2 
