IGO 
MR. L. X. G. FILOX OX THE ELASTIC EQUILIBRIUM OF 
In like manner it can be shown that 
w = Wr.z + S ( — a. 
7 — 1 ^ 27 ^ sin I'z 
47 — 1 a 47 — 1 / Jc 
27 — 1 p {rr),^„ , 7 
_ _ f 
47 - 1 J 
0 yU 47 — I j 0 At 
^ ck 
(«). 
^ (5 ^ li'=a A T (/' 
3 a + '2fx fjL 3 a + 2fj, fi 
= s~dz 
Jo 
s. l^eing the stretch parallel to the axis in a cylinder which is under a tension Q 
across its plane faces and a radial tension {rr),.^a- 
Thus the longitudinal and radial displacements are, to a first approximation, the 
same as if the cylinder were su23posed made up of any number of thin circular 
laminae, piled up on top of each other, the longitudinal tension in any lamina being 
unifoim and giving a total tension equal to the total pidl of all the external forces 
acting on the cylinder on one side of the section considered. 
Further, the shearing stress rz at a point inside is found to the same approxima¬ 
tion to be given by 
= 2 (/i'?') -- sm kz = 2o„ sin kz = - —^— 
fjb 01. a a jjb 
so that, in tlie parts of the cylinder to which external shearing stress is applied, and 
in these only, there is shearing stress inside the cylinder, Avhich shearing stress is 
proportional to the distance from the axis. 
The other stresses, rr, 0^, zz, are found to the same approximation to be all 
constants for any given value of 'i. 
22 = 2g2 cos kz = |V (z) dz = Q, 
rr = (j)(f, = iMf{z). 
It follows from the aboAm that the action of any radial pressure Avill be purely local, 
and also that, AvhateA’er the manner in Avhich the cylinder is “ gripped ” and the pull 
is applied, the stress in the portions of the bar betAveen the points of application of 
the pull reduces practically to a uniform tension. 
The above results are soineAAdiat remarkable as tending to shoAv hoAv A'ery restricted 
is the effect of local stresses, proAuded they leave no total resultant, and hoAv, Avhen 
