1888.] to the Collapse of a long thin pipe under pressure. 289 
The right-hand side is least when n=1. Hence the critical value 
of the thrust is 
T = EIn’*/?, 
as on Euler’s theory. 
(2) When we apply the same principle to the stability of a 
tube, we must take the same forms for the displacements as those 
required in treating the vibrations of a ring or cylinder given 
by Lord Rayleigh in his Theory of Sound, Vol. 1, § 233. We 
suppose the cylinder to be infinitely long, of circular section, 
radius a, and thickness 24 small compared with a. Let P be 
the constant external hydrostatic pressure, 7’ the thrust across 
apy generating line per unit length so that 
Pay: 
Let the potential energy of bending of the element ds of 
the circular section per unit length of cylinder be 
2Bds {0 (1/p)}”, 
so that B = 2h’ EH/(1 —o”°), 
where # is Young's modulus, and o is Poisson’s ratio. 
Suppose the system slightly displaced so that the point on 
the circumference whose cylindrical (polar) co-ordinates were 
originally (a, @) is displaced to (7, @) where 
r=a+6ér, 6=0+4 00. 
If e be the extension of the element of arc ds whose original 
length was adé@, we have 
(ad@)° (1 + e)’ = (ds)’ =(dbr)’ + (a+ dry (dé + doe)’, 
at i al ér\? =) 
so that (1+e) aor i +(1+7) fase ; 
Hence to the second order of small quantities 
5 A (AIO, 9 (Br dBO) , (bx dBON, Br ds 
e+ = (Gp 2 (7 + a temas) 2a a 
Now the displacement must be one in which the extension of 
the surface vanishes to the first order of small quantities*, thus 
or ms dod A 
G=~ dO 
while to the second order 
ele a 9 or de 
cay a a ah alOke 
* “On the Stability of Elastic Systems,” p. 210. 
Ofsgoth aetna eer (2), 
