1888.] to the Collapse of a long thin pipe under pressure. 291 
The circular form will be stable provided that the potential 
energy is a true minimum; for this to be the case the total in- 
crease of potential energy must be positive for any displacement, 
so that 
6V +5W,+6W,>0. 
Hence 
vy - > (n? — 1) (A,? + B,”) —dre°PX (4,7 + B,?) 
— drraTd (nv? — 2) (A? + B?) > 0, 
whence substituting the value Pa for 7, we find 
> {B/a*. (n? — 1)? — P (vn? —1)} (4," + B,”) > 0. 
If the displacement be such that all the A’s and B’s vanish 
with the exception of A,, B,, the condition of stability will be 
JOG IDG Hel, 
, lor 15 
or P<2(n ie ae 
If P is greater, then the cylinder will become unstable and 
will collapse into segments, the number of curved arcs being 2n. 
Unwin’s formula gives in our notation for the collapsing 
pressure 
h? 
P=27v7kK —. 
3 a 
The displacements for which n=1 being only motions of 
translation of the whole cylinder, the least collapsing pressure 
corresponds to n = 2 and will be 
Jo Ns LE /thickness\° 
det wer ager! 
By sufficiently diminishing the ratio of the thickness to the 
diameter we may make the strains produced in the cylinder by 
this pressure as small as we please. For if f be the compression 
of the substance along the circular sections of the cylinder (so 
that its circumference is diminished in the ratio of 1—/f to 1), 
then 
PAE adi aces a 2h 
1-—o 
whence taking the value of P just found 
thickn ss) 
diameter 
