PLANE STRAIN IN BIPOLAR CO-ORDINATES. 
285 
(3) If the centre distance is less than half the external radius, we have, in addition 
to the maximum tension (52), a minimum at /3 — 0 given by (54). There are 
no other maxima or minima and the stress decreases steadily from its value at 
the thinnest part of the cylinder to its value at the thickest part. 
On the internal surface we have 
/3/3 = —P 1 + 4P 1 M (cosh a! —cos/3) {sinh {a x — a,) cos /3 + sinh a x cosh (a, — a 2 )}, 
or, expressed in terms of the radii and centre distance, 
^ _ p , 2 P 1 r 2 2 { (r 2 2 - d 2 ) 2 -r 2 (r x + 2d cos /3 ) 2 
] (r 1 2 +r.f){r 2 -(r 1 ~d) 2 }{r 2 -(r l + d) 
. . . (55) 
Hence it may be shown that 
(l) If the centre distance is greater than one-half the internal radius the maximum 
stress in the internal surface occurs at the points corresponding to 
cos /3 = — rj2d and is 
-Pi + 
2 P : r 2 2 (r.f—d 2 ) 2 
(56) 
{r 2 + r 2 2 ){r 2 2 -(r 1 -d) 2 } {r 2 2 —(r 1 + d) 2 } 
( 2 ) If the centre distance is less than one-half the internal radius the maximum 
stress is at /3 = 7r, i.e., on the line of centres at the thinnest part of the cylinder. 
It is 
-Pi + 
2 Pir 2 2 (r 2 + r 2 — 2 i\d — d 2 ) 
(r 2 +r 2 ) (r 2 —r 1 2 —2r 1 d—d 2 ) 
(57) 
(3) The minimum stress is at 6 — 0 , the point where the line of centres meets the 
internal boundary at the thickest part of the cylinder. It is 
_p , 2P : r 2 2 (r 2 2 + r 2 + 2 r x d - d 2 ) , , 
1 (r 2 + r 2 ) ( r 2 — r 2 + 2 7\d — d 2 ) 
This may be shown to be essentially positive if P is positive so that, as would be 
expected, the internal boundary is everywhere in a state of tension. 
A Cylinder under Externcd Pressure. 
Putting Pj = 0 in (50) and (5l) we have on the internal surface 
/3/3 = — 4P 2 M (cosh dj — cos/3) {sinh (04 — a 2 ) cos ^ + sinh a. x cosh (a x —a 2 )} 
2 P 2 r 2 2 {(r 2 2 —d 2 ) 2 —r 1 2 (r 1 + 2d cos ^) 2 } / 5q \ 
{r?+ri){r 2 2 -(r x -d) 2 } {^-(n + d) 2 } . 
and on the external surface 
/ 3/3 = — P 2 + 4 P 2 M (cosh a 2 —cos /3) {sinh (aj —a 2 ) cos /3 —sinh a 2 cosh (dj — a 2 )| 
= -P, - 
2P 2 r x 2 {r 2 (r 2 —2d cos /3 ) 2 — (r 2 —d 2 ) 2 } 
(60) 
