ness, and as the circumferintial tension also tends to reduce
the thickness, /^ is necessarily less than /, and the fraction

(^'i - r) -{I- fi) jggg ^^^^ ^Yie fraction 'J^'. Hence the

r + i r

tension on the fibres of the external circumference is less
than that on those of the internal circumference, and the
former do not take their fair proportion of the work of
resisting the disruptive effect of the internal pressure.*
Lame was the first writer to accurately determine the
effect of this inequality of stress throughout the thickness
of the cylinder on the supposition of extension being
directly proportional to stress. He obtained the formula

f=— — ', where /is the tension at the internal cir-

cumference, P the internal pressure, R the external radius,
and r the internal radius. We have omitted from the
formula the term involving the external pressure, since, in
such cases as we are concerned with, the external pressure
will, in general, be comparatively very small. The steps by
which this result is arrived at may be consulted in Lame's
" Traite de I'Elasticite," or Ibbetson's " Theory of Elasticity,"
or Rankine's " Applied Mechanics,'' the result obtained
being the same in each.
P R- + r-

The formula may also be put m this form : /=.
